What convinces them that the Christian god is real?
His tangible presence. There is a lot of christianity that is solely philosophical and intellect based. That's not going to win many people at all. However, when you encounter the spiritual side... It becomes very hard to deny.
Basically, the gifts of the Holy Spirit in action. I mean... That's what we're told they're there for. "As a sign to unbelievers" 1 Corinthians 14:24-25.
Note BTW that prophecy doesn't just mean prediction to the future. It can include that because God is outside of time. Prophecy actually just means to speak as the mouthpiece of God.
So, when someone comes to my church who I've never met in my life and i get a sense from the Holy Spirit that this person is an accountant who has become depressed from the monotony of their Job, and I go and share that with them, find out what i heard was exactly right and I offer prayer and community to help deal with the issue... They often forget about the feeble human intellect arguments and give in to the power of God.
How do you think places like Hillsong/Planetshakers etc get so big? Other churches try to copy like the music, the speaking style etc but it doesn't work. Why? Because those things are simply positive symptoms of a church that has the power of the Holy Spirit in it. Those kind of things are all nice and good and what not but they don't matter if they're done without the Holy Spirit.
Take me for example, I was getting drunk and sleeping with whoever i could for a while in my late teens/early 20s. I went to church with my sister (very close friends) one weekend and a guest speaker from Planetshakers was there and he called me out in the service and gave me one of these "words of knowledge". Somehow he had know that I'd made a decision for Jesus early in my life, had since turned away and that I was living for myself desires. He told me my best friend would betray me that week but if I took the hit and loved my friend back as opposed to taking revenge it would all work out for the best. It all turned out exactly as he had described. My friend tried to pin a small crime he had committed on me. It sucked but I Put my faith in Jesus, took the advice of Pastor Matt Fielder and It solved itself in a way that blew my mind. haven't doubted God's existence ever since. Have had the Holy Spirit gave me plenty of words of knowledge for others since that time. I can't control it. I don't understand it. But it happens.
The more I get into my spiritual disciplines like fasting and praying the more it happens too.