I think there are two aspects to this.
1) I think that children spend too much time online, and in extreme cases they may lose the ability to communicate with real people who have real feelings. Instead communication becomes about scoring points and looking cool (that happens IRL too but is facilitated by a screen). I would quite severely restrict internet access under the age of 16 - I think there is no real necessity for social media access at that age. (It would have the added benefit of curtailing illegal (underage) access to online porn, which I think may be causing some problems too.)
2) I think that schools and school environments do not teach children how to socialise and communicate in a positive way with each other or with adults. It may sound extreme but I think there should be an assessed course in people skills, which could include aspects of communication (truth, humour, empathy, tact, use of language) and practical social dynamics (peer relationships, friendship, dating, social problem solving and parenting as well as working relationships and collaboration). (Interestingly, I think all this is basically applied English, and you and I may previously have learned it partly by reading lots of books...)
I think that schools should select those pupils who are popular (= accepted leaders) AND naturally good communicators (easy-going with good sense of tact and humour), give them special training and let them act as "seeds" for positive peer pressure - mainly diffusing situations before bullying even arises, nipping it in the bud. I think a popular kid using humour to reduce social friction between a wannabe bully and his target could be very effective, at least more so than lecturing from a teacher or other adults who only have enforced authority over children - I think authority has to be natural for any anti-bullying strategy to work. Finally, once kids have learned how to be considerate in real life, they can do it online too.
There is also something to be said for developing a thicker skin though, and I think the internet definitely helped me with that, so it's not all bad.