While I agree with you about those specific outlets, I also have no doubt that any powers introduced to restrict them would eventually be turned against the liberal-leaning media as well.
Doesn't mean it's incorrect, though. Doing something is tricky, for sure, but doing nothing is fatal.
Once you have the state deciding and enforcing what is and isn't true, any hope for a liberal democracy is gone.
Once you give the state the power to arrest people for spreading "verifiable lies", you now have to determine what "verifiable" means and who does the verifying. It's inevitably going to change a lot depending on the political beliefs of those in power.
It's the halting problem, "who watches the watchers?", etc. There's no way to have the state determining what's true and what's a lie, even in the most seemingly obvious cases, while maintaining freedom of speech and a free society more generally.
Apologies though if you feel I've misrepresented your position.