Extremism is a loaded word that has no real concrete meaning. Therefore, it can mean whatever the opinion makers want it to mean. And you can justify all sorts of abuse and torture with that sort of propaganda.
This game has been played many times in human history and it never ends well.
Murder, arson, kidnapping? Hate speech, harassment, cyberstalking?
Like in the 1940s, when we failed to take extremism seriously and 6+ million were killed?
I can find probably millions of people that would agree that they are in fact, extremists. In fact, the Nazis argued that Jews and Communists were evil. Of course, in Soviet Russia and China, the anti-communists were the extremists...and died by the tens of millions as a result.
To the extent that laws are actually broken, those crimes should be prosecuted.
Speech is a natural right and needs to be protected as such.
The U.S. went through the 40s all the while generally protecting the first amendment. The German Nazis went around torturing and bullying everyone who disagreed with them. Your proposal is closer to Nazism than anything I've heard of.
People should be allowed to say what they want, BUT - If anyone thinks that they have to say or think that jews/black people/woman are evil, less worth etc., they can fuck right off. In my opinion there's no justifiable reason whatsoever to be racist, xenophobic and the like.