Many arguments are made with minimal understanding, or based on false premises. Simply asking for more detail and forcing someone to take you through their thinking step by step can expose this.
It’s not enough just to give evidence that something is false. To convince the other party, provide an alternative explanation to fill the gap (lawyers do this when they point to an alternative suspect in a trial).
If you attack someone’s entire ideology in one discussion, their defences go up. Try to disentangle their various beliefs, or show how changing one might support others.
In general, people are much more rational and willing to own up to the limits of their knowledge if they are treated with respect and kindness in an argument.
Encourage your opponent to view the argument from another’s perspective – a stranger, or a person from another country if the argument is political. This can make them more receptive to the facts.