Quarter Finals will be 1 leg(match), semi finals will be 2 legs (matches).
Fenerbahçe will easily beat Kayserispor in my opinion. Kayseri has some talented players although they are not that well performing as a team. Ankaragücü is far weaker in squad against Trabzonspor.
Other 2 matches are hard/tough. Hard to say Galatasaray will easily beat Başakşehir. Başakşehir is like 4th/5th stronger team in league with sort of good coach. Sivas and Gaziantep are rather close to each other in skills so, result is hard to guess. Matches may end up in draw and extra time may happen in both matches.
I think the same for Fenerbahçe and Trabzonspor. Their opponents are quite weak and will win these matches without difficulty.
I think Galatasaray will go to the next round. Başakşehir is not a bad team, but Galatasaray is in great shape these days and there is no reason why they should not beat their opponent.
The match that will be the hardest to predict will be between Sivasspor and Gaziantep. I feel like Sivasspor has a better chance of winning. I might even be happy if they win this trophy