$15 a lesson is expensive? Woah! I'm paying $30, and that's the cheapest in the area!
If your safety is at stake, you really need to talk to your instructor and your parents. You are much better off learning on progressively faster horses-- not put on a green horse. Though insurance policies are pretty tight, most do not cover negligence, and your barn could be sued if she knows the horse to be dangerous.
Say that you want to 'stick with' your old barn. I'm sure that your parents would agree that a child who is flip-floppy is MUCH better than a child who is dead or stuck in a wheelchair!