I am a teen as well, and understand were you are coming from. I am in the "fuffy"(LOVE that) department. I excercise, but not alot. I have to bump it up now because of lifting cherries, I strained my back, and probably 20 more minutes, would have slipted it.(There is a point to this, stick with me)
ANYWAYS, my family has a history of poor backs, and my mom has been so since her mid twentys. She has been told and is living proof that strong core muscles is good support for your back, especially the abdomen. She swims, the best excercise you can do for an over all work out she believes (She swims lengths as well, not just around the pool for fun).
Maybe you could start swimming, my mom noticed a difference after she started swimming in all of her muscles; she swims 3 times a week for about 1 1/2 miles. I intend to start swimming in 2 weeks once I know the possiblity of bringing back the injury is gone.
And as everyone said, you are not fat!!!!!!!!!!!! I saw the photos and couldn't believe you thought you were fat, you are very pretty. The first thing I said is "I wish my tunder thighs were that small!!"