I'm not sure it's anything to do with the internet. A lot of this stuff is taking place offline - shouting people down at lectures and things like that.
My own theory is it's due to the decline of religion. I say that as a non religious person who has lately started wondering if I'm so right to be that way (I never used to question the righteousness of scientific atheism).
Religion can be defined as a moral framework. We have agency and we must act. Religion helps answer the question "how must I act" in a thorough and formalised way: it tells us what right and wrong is, and it does so from a position of pseudo-authority (God says it, so it must be moral).
The common thread between all these conservatives-get-censored stories we're seeing is that leadership, whether it be corporate political or academic, is staggeringly weak in the face of moral outrage. Even when that outrage is coming from people who are barely adults, people far more senior and more experienced than them fold immediately - even if they actually disagree.
I think this is because without any kind of rigorous moral framework, people don't have any footing on which to push back. Someone pops up and says "that person is the devil because they don't respect women" or whatever, and maybe that's a total pastiche of their position but it's a position anchored in the moral framework of identity politics/femininism. Without any equivalent grounding in an alternative moral framework, the leadership doesn't know how to respond, they are desperate to see themselves as a "good person" and so they're left scrambling to obey the demands of the assembled mob. If they already felt like they were good people, they'd be much more capable of resisting such attacks.
Organised religion crops up in every human society throughout history except this one - I believe we're really the first to be widely atheist. But it seems all we've done is replace one set of gods with another: now instead of worshipping invisible deities of peace, love, war, etc, we worship at the deities of minority intersections ... and crusade against heretics who deny the sanctity of that worship.