How can I get out of a karate contract if I think that it has been breached?

This may be very difficult to get out of on grounds that they are not living up to their promise to improve self-esteem, since that is such a subjective measure. In essence, the courts have difficulty with contractual "terms" which impose unquantifiable measures or benchmarks. You would probably have better luck by arguing that the karate school is both making your performance impossible--by making it so your son doesn't want to attend--and moreover, is violating the covenant (or duty) of good faith and fair dealing. That is the requirement that both parties to a contact demonstrate good faith in both negotiating and executing the agreement's terms, and sufficiently bad faith can give the other party the right to terminate the agreement. However, these arguments, too, are subjective--it's not like you can point to something concrete and easily quantifiable, like the school being closed 5 of 7 days, when it was supposed instead to be open 5 of 7--so you need to be prepared that the school may choose to fight and it may be difficult to prove your case in court. You thus may wish to consider settling--maybe pay another 3 - 6 months and they let you out of the contract.

On a different note: I've practiced karate for 25 years, and my 7-year-old daughter is currently enrolled in a wonderful school with me. I can say definitively that you are right in that no sensei or instructors should mentally abuse a child, ever. Regardless of the legal issues, you appear to be doing the right thing for your child.