What to get a teenage girl? Well, first of all you want to ask them what they want. They're at an age when details matter a LOT.
If they have a clear birthday wish, just follow their suggestion. They'll thank you for it, though perhaps not in any way you will notice. They are teenagers after all...
Seriously: a teen wants what they want - finding out what they want is a big deal at this age. They're finding they're independence.
Then again, there are gifts that can hardly go wrong. Here are a few likely suggestions, if simply following their hobbies and music preferences doesn't work.