In short, bullies need to face punishments with teeth. And the bullied need to have access to a justice system they can trust to actually work. Something like this: http://www.sudval.org/05_onepersononevote.html (scroll down about halfway to "When you were young", A True Story).
Look, our justice system as adults distinguishes between aggression and self defense. But for some reason, we treat conflict between kids as if it's everybody's fault, and everything's okay once it stops. That's unjust, and is the root cause: adults that treat order and convenience as more important than justice.
When I was a kid, I remember complaining that my (three years younger) sister hit me. The first thing my mom asked? "What did you do to make her do that?" Well, of course I answered, "Nothing," but it plainly wasn't true. Who picks a fight with someone twice their size? And over the course of the discussion it came out that I had taken her toy and teased her, and you know what? My sister got a free pass. I got in trouble.
How about that old fashioned rule, "If you can't say something nice, don't say nothing at all?" How about, if I catch you insulting or belittling someone, you have to be their servant for half an hour -- get their water, sharpen their pencils, that sort of thing?
How about a justice system that treats allegations of teasing and insulting as serious matters, and investigates to get to the bottom of things? How about a system that is concerned about kids' antisocial behavior, not because it is currently bothering the adults, but because of the effects it has on both victim and aggressor?
How about a system that punished physical violence harshly if it was aggressive, and leniently or not at all if it's defensive? How about a system that says, "If you hit back with the aim of making the other kid stop, and you stop as soon as he does, we'll hold you blameless?" You know, the same deal adults get. (By popular judgement, anyway, if not always in court.)
How about a system that responded to kids' experiments in evil with love and affection and attention, encouraging them to delve deeper and teaching them that behaving cruelly toward others is a great way to get all the stuff they want?
Never mind on that last one.