We're not supposed to believe it. The point is that Luthor believes it and that belief drives his vendetta. And the point of this episode was to show how a person in power can use the media to advance a self-serving lie if media personalities like Godfrey prioritize a "both sides" mentality over actual fact-checking. (Which is why it was a mistake for Lois to accept a debate on their terms, because that gave them control of how the narrative was framed.)
The show is making it appear that many people (maybe not the majority but enough of a percentage to ruin or damage their reputation) do believe lex's story.
Simply put reporters do make mistakes. Why isn't there press that Lois is the one who got Luthor freed. He's playing it like she did this fraud against him to keep him imprisoned.
That logic falls apart when she is literally the reason that he is now freed. She is the one who released evidence freeing Luthor. Yet we don't see that even being addressed.
I mean what do people think Lois changed her opinion on the threat level of Lex? She has believed him to be a master criminal and an extreme threat to people on the planet. Do people believe she got him released because she believes he was reformed in prison?
Now sure Luthor is obsessive. I can understand him. But not the bulk of the general public.
Now Clark is not going to do something illegal. But just in this episode both characters have multiple means of having proof. Kansas the setting of Smallville and Metropolis is a state with the law that only one party needs to know a recording is being made. Lois was in position to record multiple people who state Lex's actions including Lex himself. Superman and Clark have also been in contact where they have stated things that don't make them come across as innocent. So even if you think Clark wouldn't you feel that Lois wouldn't. Part of this has occurred in her business, By multiple people.
Then of course, we use parallels with politics and media where we have people who frequently openly lie and are believed by millions. But that metaphor doesn't hold here.
Superman has been around for probably close to 20 years by this point. Kins should now be close to 17. Give Lois and Clark a few years from when Superman first showed up to them dating, getting married and having kids who are now near 17.
This is a guy that has been saving people in small numbers and on a planetary scale for decades. Someone who gets no monetary reward for his service (the DOD didn't pay him), He hasn't seeked political power. He doesn't profit from merchandising, nor does he try and stop people from making merchandise. And he hasn't used his voice to enact political or societal change (per Bruno last season). His family isn't in private school (even in metropolis). The only thing you could now accuse him of his gaining success as a print reporter due to his abilities as Superman. A job based on his position that probably earned based on US (not even Kansas wages), likely 70 to 100 grand a year. That's hardly getting rich off of his abilities. And of course records would show he didn't enrich his parents life financially. And then his very significant pay cut living in Smalville.
Now there would be handful of people who think Superman is a hoax. And there would be people who due to his half brothers actions are going to fear what Superman could do if he went rouge. And a larger but still rather small percentage who would just like to see someone famous and well regarded take a fall.
But that wouldn't be a very significant portion of the populace.
Especially since literally a just more than a year ago, every single being on earth was saved by Superman in a very, very visible way. This isn't like he took down a nuke that no one realized was gong to blow. The effects of the transfer between worlds was happening all over the planet. Every single conscious person would have seen and felt the impact of it.
And the idea that Lex could use the media in a few months period of time since his release to enact such sweeping changes in the general population is utterly irrational.
Its poor storytelling.