Quick definition of a bildungsroman, courtesy of dictionary.com: A novel whose principal subject is the moral, psychological, and intellectual development of a usually youthful main character.
So you have a teenaged protagonist, and you’re writing hell-bent for leather to make your fantasy the story of her (or him, as the case may be) growing up and finding herself. So how do you make this old-as-the-hills plotline shine?

1) Begin as close as possible to the point where your character is ready for a change.

It’s necessary to introduce her and establish some background, of course. What has her life been like up to this point? What are her likes and dislikes? What is she rebelling against? (And of course you have to answer that question, because what is a teenage-protagonist fantasy story without the teenager rebelling against something?)
But you don’t need two hundred pages of this. It may be fascinating for you, and perhaps you have the skill to make it fascinating for your audience…but I doubt it. Even an author whose teenage protagonists I liked, Tad Williams, loses a lot of readers in The Dragonbone Chair by taking more than two hundred pages to get his awkward, innocent teenage boy, Simon, out of the damn castle. Once he does, many exciting and genuinely scary things happen, often right on top of each other, and Simon can really start changing, since the environment of the castle kept him static. But Williams is also interested in exploring the history of the castle, the Hayholt, and how Simon climbs around in its old buildings, and how Simon flirts with the scullery maids, and a good chunk of the world’s history, and Simon’s “apprenticeship” with the resident castle scholar. It rambles, and rambles, and rambles. It’s good for the world-building aspect, and for things that come into play in the third book of the trilogy, but it’s hard on the story’s bildungsroman aspect.
Any character-driven story should begin as close as possible to the character’s change of heart, or the events that inspire it. That’s common advice, and a lot of people do manage it with adult protagonists. Writers of adolescent-centered fantasy, however, often seem to assume that what their readers really want to read about is how the adolescent gets teased and bullied, or what she has for breakfast every morning, or the duets she sings with Mr. Bluebird. No, most readers don’t. We want to see how this shy/awkward/innocent/clumsy/picked-on person is going to become brave/self-confident/wise/graceful/badass. That should receive more space in your story than the flashbacks. If there is information about the protagonist’s childhood and early life that it’s absolutely essential for us to know, work it in later, instead of trying to pile it on all at once.

2) Don’t take your story with complete and utter seriousness.

The other fantasy writer whose teenage protagonists I’ve genuinely liked as people and thought worked in the story is Dave Duncan, author of the A Man of His Word quartet. (Magic Casement, Faery Lands Forlorn, Perilous Seas, and Emperor and Clown are the titles—and why, yes, those all do come from the same verse in Keats’s “Ode to a Nightingale”). He uses what could be absolutely stereotypical characters, Inosolan the princess and Rap the stableboy, who’s her friend and protector. And from the beginning, he refuses to treat them as the center of the universe. They have to grow and change to justify even being treated seriously. Inos gets into all sorts of ridiculous situations that are her own fault, and gets scolded by both her father and her aunt Kade (one of the most wonderful characters in fantasy). Rap has what seems to be a magical ability to control horses, but it’s used for absolutely practical purposes at first, and then Rap goes up against several enemies with whom his magic helps him not at all. Throughout it, the author gently pokes fun at them, and the other characters treat them as ordinary people, or, at best, as political and magical pawns.
And by the end, it all works.
Other fantasy authors can do, and have done, far worse than imitate this way of looking at their protagonists. Step away from them for just one blazing second. Think of them in the context of the whole damn world, and if that doesn’t work—because they really are the most important people in the world, or whatever—then think of them through the eyes of people who just don’t care. That can serve to knock the characters down a peg or three in the author’s mind, which itself, rather than the way the characters behave in the story, is often the problem. Don’t insist that your readers regard the teenagers with holy awe. It doesn’t work, and it irritates people. And most of the time it’s not even fair, since these are, and are meant to be, typical teenagers, not shining saints.

3) Give them lessons in the school of hard knocks.

Too many teenage-centered fantasies disgust me because the protagonist doesn’t have to do a goddamn thing to get their happy ending. It’s a stroke of fortune, a stroke of “love” on the part of someone else, or the protagonist winning because of bloodline or magic. The deus ex machina endings I ranted about earlier are more common here than anywhere else in fantasy. Even Williams’s trilogy suffers from it to an extent; I still liked the books, but the ending dimmed my enjoyment.
Your adolescent may respond to problems and situations and people around them in a typically adolescent way, by hiding her head in the sand and refusing to deal with it, but sooner or later you need to dig her out and put her on the road. She needs to learn responsibility, self-control, wisdom, perceptiveness, inner strength, and how not to bury her head in the sand.
Too many fantasies do nothing to teach their protagonists that, quest for self-realization or not. However, at least with an adult protagonist who is recovering from losing a lost love or something like that, there is a sense that this character might have learned something in the past. An adolescent who shyly crushes on a teenage boy, takes lessons in magic, and defeats the villain in the end with the power of Love has learned—what? That magic is the answer? That love is always returned? I don’t know. Certainly not what the author seems to set her out to learn.

4) Suffering by itself is not the answer.

Sometimes people go the opposite route and make their teenage protagonists the center of an angst black hole. Their parents hate them. (I always feel sorry for the parents trapped in the story with these Brats From Hell). They have [insert Teen Issue]. They have no friends, and everyone near their age teases them. Their siblings are more beautiful or accomplished than they are. And so on.
Then by the end of the story they’ve “grown up” because “they suffered.”
Get over it. An adult is perfectly capable of suffering for years without learning how to stop making the same mistakes. I think a teenager is actually more capable of it, since he or she doesn’t have as much hindsight or experience to draw on, and is seen as someone whose mistakes aren’t really her fault. This is fine if you do intend to write a book where the teenager isn’t really a hero and/or has only partially finished growing up by the end. It isn’t okay if the teenager really is supposed to be growing wiser and better as the book goes on.
If she suffers, remember that her reaction to the suffering is the most important thing. Suffering doesn’t exclude you from having to do characterization. It makes it harder, drives you deeper into your protagonist, and should make an adolescent into an adult faster most of the time. Don’t skip over the steps in between.

5) Make the epiphanies genuinely startling.

It is so very easy to turn a quest for self-knowledge or self-realization into a quest for plot coupons, indistinguishable from the normal fantasy one except that the hero’s finding platitudes instead of people or objects he needs to save the world. “Oh, here’s where he learns that love is always the answer. And here’s racial tolerance. And here’s where he learns that his people were wrong for persecuting the elves. And here’s the lesson about the Mother Goddess’s religion being the one true one.” Yawn. The author lays the themes on with a trowel, and the story drowns in Meaning.
Be more subtle than that. Present the teenage protagonist with at least a few situations where there’s no easy answer, and even though he makes a decision and lives with it, he’s not a happy camper. Lessons that are easy to embrace, on the lines of “Why can’t we just get along?” don’t challenge him. He’ll trundle from plot coupon to plot coupon without much trouble, and the audience will invest in toothpicks to keep their eyes open.
Also, have the teenage protagonist learn a few unattractive or troubling things about himself. So he might be the savior of the world and the last heir of the king’s bloodline. But he might also have been thoughtlessly cruel in the past, or genuinely and persistently wrong about someone, or so concerned with his mysterious heritage that he failed to appreciate what he had in the present. Let his face flush, his hands tremble, his eyes fall. Frustration and shame can be great catalysts to growing up.

6) Don’t treat the character’s beliefs and experiences as a tabula rasa.

Those childhoods that too many authors spend too much time on? Often, they’re lacking in core principles. The adolescent who emerges may have some strange magic, a mysterious past, and abuse. And too often, that’s it. The author leaves her a blank slate so that the experiences she’ll have in the outside world impact on her as hard as a child’s first lessons. That’s why it’s so easy for her to wander from platitude to platitude and learn “new things” about herself. She doesn’t have a whole lot of old things to really change.
I’m sorry, but unless your character has amnesia or is mentally handicapped, this is a frank cheat. A teenager is not a child. If you’re using human characters with human-like psychology, then their childhood training can and should bind them, help them, and hinder them. They shouldn’t be able to go about picking up new languages and sword-skills like nobody’s business, and they shouldn’t be absolutely free of embarrassing beliefs.
Some authors try to sidestep this by showing the adolescents making factual mistakes. “Oh, well, she never knew that elves were persecuted, so she’ll have to learn that they were!” But the moment that the adolescent learns the truth, she abandons her people’s ohmygod so prejudiced beliefs and adopts the new ones with a gulp. No crisis of faith, no asking why her parents believe that, no insinuation even that it was a mistake on the humans’ part and not a deliberate lie. She just jumps headlong into the new, and right, and righteous, belief system.
People whose perceptions are exactly the same as the world around them, who are always, always right about good and evil even though they may have some history wrong, are canon Mary Sues. They’re perfect. They can’t be really wrong. They have nothing deadly or bigoted to unlearn. They’ll get over the consequences of any factual mistakes soon enough. And why? Because the authors write them as blank slates that can get righteousness imprinted on them.
This defeats the whole point of a growing-up story. The character has to have something to grow up from, not merely into.
The bildungsroman is one of those stories that I love seeing in the hands of a competent author, but too often its handlers are ham-fisted.