I've seen the series and read the first book and thought it did a pretty good job. What do you feel the TV show got wrong- not being argumentative btw, I'm just interested!
In the show they made her out to be... American. Rude, spoiled, etc. Other than that I just wish they had the time and the audience to tell the FULL story in all it's glory. Of course it's expected that the sub-plots have to be trimmed of the minutiae, but the complexity and ambiguity is part of what makes ASoIaF so amazing.
Using the common motive of "making the show appeal to a wider [American] audience," it seemed to me that they wanted her character to hit closer to home than a polished young lady would. Of course, my opinion is just a shallow one - a thought in passing.
Thinking on it now you are probably right to question me on my association of 'rude and spoiled' with 'American,' but that was just my gut reaction. FWIW, I'm an American with minimal knowledge of the demeanor teenage girls from other societies, Western or otherwise.