Galatasaray is playing really badly in the league this season. As you said, it is not normal for a team that plays so well in European games to be among the worst teams in the league.
With the change of coach, I think Galatasaray will return to normal in a few games. They could finish the league in tenth place or a similar result this season. This has been the worst season they've had in years
They have lost the last 4 matches in a row in Galatasaray. They have conceded 10 goals in these 4 matches. This suggests that the main problem in Galatasaray is a weak defense. Unless new talented players are added to the defense, their game will not improve.
Besiktas was out of the top 10 of the points table even a few weeks ago. But now they are the sixth team in the table. And if they win the next match, maybe they can go to the third place in the table. Galatasaray still has plenty of time. If they can strengthen their defense, maybe they will be in the top 10 of the table.
There are 38 games right? If I am not wrong that means there is 15 more games left, 45 possible points. Even if they could get like 30 of those, that is going to be midtable at the very worst case for them. It means around 2 points for each 3 point possibility, a lot of wins, a bit of draws and very few losses. Could they do that? They potentially could, if it was all about the roster then they wouldn't be able to go to next stage in europa, they are decent enough to go to next stage so they are capable of doing better in the league.
The key word here is "capable", just because they are capable doesn't mean they will be able to do it. Just realize that they are not really doing something that they should be doing right now, and that shows that they suck at the league so they could also keep doing bad in the league as well. They may not relegate, but they do not feel like they will recover too much neither.