Liberals have found an interesting rationalization that allows reconciling support for free speech with the need to censor ideas outside that threaten liberal ideology.
The trick is to claim that the total sum of valid ideas falls within the liberal ideology. All the ideas that are contrary are therefore fundamentally invalid, and thus can be treated simply as noise.
This is why liberals love the fake news and disinformation narrative so much. In their mind, they’re not censoring valid ideas that are contrary to their own ideology, but are rather fighting against noise that has no fundamental value. Since these aren’t valid ideas to begin with, liberals don’t see censoring them as a form of censorship.
On a related note, I’ve also noticed that we basically have a Darwinian competition between different world models. Different groups of people subscribe to a particular explanation of how the world works, and that becomes their ideology. Some people go through this process consciously, but vast majority just internalize the world view from people around them as they grow up.
What’s interesting is that once somebody settles on a particular world view then it’s natural for them to reject conflicting views since there’s no way to prove whether one view or the other is correct in most cases. And I tend to think of this in terms of thermodynamics where people have complex graphs of ideas in their heads, and when any particular idea is challenged then the whole set of ideas associated with it has to be reworked as well. It’s easier to simply discard conflicting ideas than to go through the process of rethinking a bunch of things you’ve internalized over many years. This is why it’s typically very hard to change people’s ideas no matter how good your argument is. The cost of integrating this new information is just too high to bother in most cases.
What typically causes people to go through this process is when they start seeing the drift between their world model and the material reality they experience. When mainstream liberals start experiencing a continued decline in their material conditions then it becomes difficult to continue believing that everything is getting better and that they’re living their best lives under the most enlightened system possible.
Hence why a lot of people started questioning things after 2008 crash, and as economic disasters continue, we see more and more people falling out of the liberal mainstream. Unfortunately, as you rightly point out, many of these people end up on the right because the right ideology is very close to liberal ideology, so it’s much easier for people to internalize those ideas.