Prove what? That Trump didn't do something? Your own link does that. One of the headlines (first link you posted) is 'FCC chair, helped take down Jimmy Kimmel with words, not actions". Just like anybody else, the FCC chair can say what he wants. That's not the same as taking legal action to force something. That's not censorship. Quoting further from the same link: "Carr did not violate the letter of the law. That’s because the First Amendment bars government action limiting free speech. And Carr didn’t take any action — he merely, and perhaps ironically, spoke."

And Trump can be as happy as he wants to be. It shouldn't be any surprise that he doesn't like Kimmel. So what? Trump expressing his opinion that ABC should cancel Kimmel's show is not censorship. Legal action to force Kimmel's show off the air would be (depending on the exact circumstances I suppose) and that has not happened. And, by the way, Kimmel's show is already back on the air (except for affiliates who are choosing not to show it which is about 25% of them last I heard).

BTW, this was the comment from Kimmel that has caused perhaps the most controversy, in no small part because it is untrue.

“We hit some new lows over the weekend with the MAGA gang desperately trying to characterize this kid who murdered Charlie Kirk as anything other than one of them"

There is speculation that Nextar, owners of some of the ABC affiliates that are still not showing Kimmel's show is making that decision because it is currently trying to merge with another media company (a $6.4 billion deal) and this needs government approval. Personally, i think further media consolidation is a bad idea anyway.

I'm not a huge fan of Trump or of cancel culture on either side. But I'm also not a fan of hyperbole and to say that ABC has been censored is certainly that.