Independent fact checkers not associated with government authorities (like what Twitter is doing in this case) seems like one solution.
I'm really bugged by the leap of logic that fact checkers will supposedly always parrot the government line, when even this specific thread itself is about a fact checker existing that's going against the president's line.
Conspiracy theories and baseless nonsense is the price you pay to be able to criticize those in power. It is a price worth paying.