I must be lucky, because I spent 9 years in private schools, and I saw no bullying of any kind. Not even serious teasing and that was in the 70's and 80's. Sure, there were some kids who were thought of as "dorks" and "geeks," and there was joking behind people's backs, but nothing really mean-spirited.
If you really, REALLY, think the average private school has the same type of law-of-the-jungle type atmosphere of the average private school, I can only say that doesn't come close to my experience, even more so when I consider my children's private school.
Easy to see why this experience made a permanent impression on our family - especially when it was the public school kids who were calling to check on her, sending cards, stuffed animals, coming by with treats, and helping her with make up work. The kids in her Sunday school class did nothing - she was told she was an unrepentant sinner.
Does this mean I don't think bullying goes on in other places? Of course it does - but this is our experience, others will vary.