Anne McCaffery's Pern series
I'd consider them with an asterisk; some parts of it don't age particularly well, though the concept remains iconic. The Harper Hall trilogy holds up a bit better, IMHO.
For dragon-related reads that might appeal to an older teen girl who likes romance and action, I'd throw in Marie Brennan's
Memoirs of Lady Trent (alternate-world lady naturalist traveling the globe to study dragons; not much romance, but plenty of adventure) or Carrie Vaughn's
Voices of Dragons (not much romance either, but a special bond between a teen girl and a young dragon in an alternate-modern world divided between dragonlands and humans), possibly Jen Williams's
The Copper Promise (a ton of adventure with a little romance, sort of an update on the classic Fafhrd and the Grey Mouser but with much better gender balance and representation - though the dragon in question in this one is an ancient and evil menace) or Elle Katharine White's
Heartstone (which is quite literally
Pride and Prejudice rewritten in a world with dragons and monsters.) Naomi Novik's Temeraire series is great (though it ends weaker than it starts), but might come across as a bit stiff or slow, especially the first volume.