The US has survived a lot of extreme speech. The Communist Part of the United States existed through the entire Cold War, and while they were sometimes hassled, they continued to publish their newspaper and pamphlets. They still have a web site. The American Nazi Party used to have a big sign on their HQ: "White Man Fight - Smash the Black Revolution Now". I walked by it as a kid. The sign stayed up until one of their members shot their leader. The Klu Klux Klan was an out and out extremist organization, but no one suggested in their heyday they should be muzzled. Just prosecuted for killing people. Even censorship of pornography mostly ended.
The pushback began in the 1980s, with the Meese Report on pornography. The authors were looking hard for something to criminalize, and hit on child pornography.[1] Not just making it, but possessing it. That was a new thing at the time.
The next big pushback was on Holocaust denial, first criminalized in Germany in 1985 and in France in 1990. A few other countries have followed, although not the US. This has gradually broadened into European laws against "hate speech", which is often political and very much a matter of opinion. While it seems to be settled law that the US government cannot prohibit hate speech, there's a lot of pressure on private organizations to do so.[2]
A new problem is overuse of the "big lie" technique.[3] For decades, nobody in a country with a free press tried that much, because it didn't work. The people pushing the lie were quickly discredited. That obstacle seems to have been overcome. That's leading to a new era of censorship merely for being wrong.
And that's the way it is.
[1] https://en.wikipedia.org/wiki/Child_pornography#History
[2] https://en.wikipedia.org/wiki/Hate_speech_in_the_United_Stat...
[3] https://en.wikipedia.org/wiki/Big_lie