How many games is the demotion prevension for going up a tier

It's based on your MMR. If your MMR is bronze level, you're going to lose more LP from losses and gain less from wins, until your league matches up with what your MMR says it should be. Conversely, if your MMR is higher, you're going to lose less from losses and gain more from wins, until you get promoted. If your MMR is high enough relative to your league (let's say you keep getting people trolling your promos) you can even skip the promos entirely.