I have a friend who was bullied so abusively and thoroughly at a private school that his family pulled him out and moved him into public school (which turned out to be a very good school system with very little bullying, fortunately for him).

Obviously that's just one data point, but I don't think the incentive structure at a private school is what you think it is. There's a lot of incentive to keep issues quiet, and a lot of pressure to fit a certain mold.