Mosques have a responsibility to reject extremism and report dangerous individuals to the FBI. And they do: https://www.theguardian.com/world/2012/mar/20/fbi-informant

A mosque in America that openly fostered extremism like 8chan did would of course get shut down.

I think you are trying to turn the above argument on it's head, but honestly I have no idea what point you are trying to make.

"The free speech of muslims"? It's almost as if you think 9/11 was caused by a domestic terrorist. You do realize the hijackers were mostly from Saudi Arabia right? And there is no free speech there? So... what are you trying to imply? Please use more clear language instead of just trying to meme.

Clearly you understood what I was trying to say, you're just acting dumb to avoid directly engaging with it.

These arguments can be directly used to argue for the suppression of Islamic religious speech. Using the same "guilt by association" reasoning, you can easily argue that 9/11 is proof that Islam itself is a hateful and dangerous ideology which, when spread, has "consequences".

The distinction in this case is that 8chan is simply a “poisoned well” which can be boarded up until the next one takes its place, whereas “Islam” is a whole ideology followed by a billion-odd people.

“Extremist Islamic terrorists” on the other hand, that is a narrowly defined enough population to feel justified in going to war against.

But the point is, no one’s speech in particular is being suppressed. To keep the metaphor, a “mosque got too radical” and was shut down. People who were practicing there will have to find another place to go.