Both sides push restrictions, but in different ways. Democrats push for stricter moderation to curb harm, usually by pressing platforms to enforce their own Terms of Service more consistently. Republicans try to strong-arm platforms into protecting their own voices - the latter is imho probably worse.
Well we heard and we've also seen first hand what some American liberals did and wanted to do.
Charlie Kirk lost his life in the ultimate act of liberal censorship.
Numerous liberals then claimed online that it was all good and that he had it coming to him, in some cases his wife and children should follow.
So all in all, I'd say liberal censorship is far worse. We can see that what these teachers and kids are taught by their liberal leaning educators is violence towards people with differing views, helped by dehumanising the other, by calling them a Nazi, fascist, bigot, KKK, transphobe, etc.
We also need to remember that this is the first year the pendulum in the US has swung right. Since Obama 1, we've had cancellation, demonetisation and censorship of mostly conservative voices online. People were fired for being conservative in their views. Now the pendulum is going right and we'll see how much the right censors the left, I suspect there will be less censorship.