The most important thing to keep in mind is that fringe beliefs like flat-eartherism, young-earth creationism [EDIT: not to be confused with creationism in general], lunar landing denialism, etc. are not subscribed to for logical reasons but emotional ones. It's a club that you can join whose rules are very clear -- all you have to do is to profess to subscribe to the belief. Once you're in you get to feel special, like you are part of a small elite group who knows a Deep Truth that no one else does. As such, no logic is going to convince them. The only thing that will extract a person from such a belief system is emotional support, something to fill to emotional void that giving up the belief and hence the membership in the group will leave.

Good luck.