One interesting (and probably good) thing about the Turkish Cup compared to the Super League is that it’s more open to surprises. If you look at the winners from recent years, you’ll see that it’s more likely for different teams to win compared to league championships. I think this is what makes the Turkish Cup important.

As for the main question, it’s really hard to make a prediction and be correct, but since I’m a Galatasaray fan, I’ll say Galatasaray.

Because so far some teams are more focused on the League and the Turkish Cup as an opportunity for some team to win the title, but still playing against the top team from the Turkish Super League is also not an easy matter for the lower team, even surprises like the ones you mentioned do not happen that often.
I agree that Galatasaray could be the team that will win the title, they are at the top in terms of the quality of their playing performance and of course some team below them will have much harder time beating Galatasaray until they actually reach the final, plus the performance of Besiktas who previously won is also now seems to be decreasing.

Fenerbahçe announced that they wouldn't participate in the Turkish Cup and the federation confirmed this. But it was rumored that Fenerbahçe had a penalty from last year, so they wouldn't be participating in the cup this season anyway. It's an interesting situation. I think all teams should be in the cup, lack of competition is not good.
It's hard to talk about favorites at the moment because the group stage is a first time format and we don't know how it will turn out. But the natural favorites are of course Beşiktaş and Galatasaray. Considering Trabzonspor is also having a bad time in the league, they might give importance to this cup.

Interesting results today. 2nd division team İskenderunspor defeated Super League team Hatayspor 1-0 and eliminated them. Another 2nd league team Kırklarelispor also eliminated Super league team Adana Demirspor by beating them 2-0. Kasımpaşa, who took the field with their U19 team, is 1-0 up against Gençlerbirliği and it looks like the match will end like this.
I wonder if this round will end as soon as possible so we can see how the group draws will be and how the new format will work. Cup games are not really enjoyable without the big teams getting involved.

It was a strange draw. For some reason, it's all working in Galatasaray's favor again. Galatasaray will play 1 easy game away and 2 home games. But Fenerbahçe and Beşiktaş will play 2 away games and 1 home game. Beşiktaş will play their easy game at home and 2 difficult away games. It's as if they want these teams to get tired in the cup, have an impact on the league and let Galatasaray become champion comfortably. With the example of Beşiktaş, who played all their matches away after the Europa League, this federation has no validity left. Football in Türkiye, with all its institutions, takes every decision in favor of Galatasaray. In the future, this period will go down in history as a black stain. I hope we can see those who caused this being tried and punished, but I don't think so.

Beşiktaş won their first and in my opinion the most difficult game of the group round and most probably secured their advancing from the group. There is no chance they will lose points in their home game against Kırklarelispor. Therefore, 6 points will be enough to be top 2 in the group. In this group, Sivasspor and Antalyaspor had a chance to advance with Beşiktaş. Sivasspor lost today and reduced their chances even more.
The game wasn't very good but we have to remember that this is a cup match and a group stage. The only aim of the players was to get 3 points and they did that. On Mario's assist, Rafa Silva hit the ball well and scored a clear goal. Other than that, their offensive variations were weak. At least they gained some morale with a win after a bad run in the league.

Başakşehir played well but had to settle for a draw. There were 4 goals, red cards, penalties, everything. Galatasaray took advantage of their large squad. But the penalty they missed in the last minutes was unfortunate for them. I think this is not a bad result for both teams. Depending on the situation of Eyüpspor and Konyaspor, this extra 1 point can even bring them the round.
Trabzonspor and Beşiktaş are currently very advantageous in their group because they beat their direct opponents in the group. All Group B matches will be played tomorrow. The new format doesn't allow for many surprises, but there could still be a big team that doesn't advance from their group.

There was a Galatasaray-Başakşehir match in the Turkish Cup today and it ended in a 2-2 draw. Mitcy Batshuayi missed a penalty in the last part of the match, Mertens could have used the penalty kick instead. Galatasaray needs to find a solution for defense during the transfer period, we see as many goals as we score.

Başakşehir also played a good match, but there were big mistakes and carelessness in Galatasaray defense, they caught it well.

Başakşehir could have won the match. They fought very well. Many people showed Galatasaray as the favorite, but Galatasaray had a hard time. Also, they missed a penalty in the last minute, Batshuayi can be blamed for this. He could not take a penalty even though he was a forward. Also, goalkeeper Muhammed Şengezer was very successful for Başakşehir. We have to say this. We drew with Galatasaray in the first match in the Turkish Cup. Frankly, I think this was a surprise.

Today, Kasımpaşa and Fenerbahçe will face each other and Fenerbahçe announced yesterday whether they will participate in the cup or not.

This is also strange. Can teams from quality leagues in Europe do something like this? I wonder if they will participate or not?

If both teams played this match with their A teams, Kasımpaşa would win and I would play. Fenerbahçe would participate but mostly as a substitute, I don't know if they will play with the U-19s. Kasımpaşa had previously said they would play with the U-19s but I don't know if they will do something like this against Fenerbahçe.

The Turkish Cup doesn't have a proper system either. Things change every year. It's like a test drive. Even this shows how far behind we are in football. There are 2 more cups in England outside the league. It would be better if they take them as an example and add 1 more cup and the winner of that cup goes to Europe. Of course, since the Super League clubs have the upper hand, no TFF president dares to do this.