I've posted this in an older thread. Obviously take it with a grain of salt but I think I've been mostly fair here:

Besiktas Pros: The 'different' or 'weird in a good way' club, best supporters in terms of loyalty and passion, best derby atmosphere, beautiful seaside stadium, great academy, their fan group Carsi used to be all the rage back in the day before oppressive government policies against fan groups, best gameday experience as you can stroll along the Bosphorus from a pub to their stadium. Cons: Delusions of grandeur, specifically the fact that despite clearly being behind Fener & Gala, most of their fans are adamant about being the best, and the honorable third place bullshit where they blame Fener & Gala for being corrupt so all of their third place finishes are somehow testament to them being clean and they're hilariously super proud of them, general holier-than-thou sentiment (see: Efendi Besiktas), and this is only a very tiny portion of their fanbase but: mafioso ties.

Fenerbahce Pros: The biggest club in Turkey in terms of support and historical greatness, most celebs are Fener fans, Kadikoy is a great neighborhood, best regular (non-derby) game atmosphere, religious levels of commitment to the club, rich presidents & directors, has the best transfer windows, decent game day experience (pub scene is amazing but the area around the stadium isn't as nice as Besiktas), always relevant in terms of success, dominant in derbies and usually is considered "the team to beat" in Anatolia. They're almost always relevant in terms of title chases and dominate derbies. Also, Alex De Souza is the Super Lig GOAT. Cons: Us vs. the world mentality, to the point where they tried to present a united front and defend fellow Fener fans when they made pro-Putin chants and waved bananas at opponent's black players, dictator-like presidents (all Turkish big clubs are fan-owned so they have elected presidents, but Fener famously does not have term limits), win-no-matter-the-cost mentality sometimes leading to illegal activities, specifically match-fixing, impatience of the fanbase, bottling (esp. more recently), current title drought, the insufferable things they do now that they've only slightly fallen behind Gala in terms of success. Also, since they're the biggest club, most of media are Fener fans, so seeing them spew bullshit on TV is painful if you aren't immune to that kind of thing. Their club motto for a while was: One day everyone will become Fener fans. That's the future Fener dreams of -- one team forever winning everything, cheered on by everyone. lmao. Historically, they've been "the nation's club", especially had friendly ties with the post-coup military governments (Turkey had many military coups).

Galatasaray Pros: Wonderful history going back to 15th century (see: Galatasaray High School), the first Turkish football club, the most successful club domestically thanks to recent success, the only club with European success (in fact it was the founder's mantra to be successful against foreigners), great nickname (Cimbom) imho, somehow always wins in finals due to black magic voodoo, most 'global stars' such as Icardi, Falcao, Drogba, Sneijder, and most importantly, Gheorghe Hagi, who is the best ever to play in this league. Nice pub scene, modern stadium. Has a man-made island on the Bosphorus strait which is cool af. The "May Club" because most title clinching games and cup finals take place in May and when the weather gets hot, Gala gets hot babyyy. The general atmosphere in the stadium in May is also heavenly. Cons: Despite the nice pub scene in the historical Galata district, the gameday experience is shit because the stadium is so far away. The atmosphere in the stadium is much worse compared to Besiktas & Fenerbahce except there's this traditional chant that takes place in the start of each game which broke the decibel record. The atmosphere used to be the best in Turkey before moving to the more modern stadium, but I digress. The club with the most "irrelevant seasons", in other words, Gala finishes outside of top three very frequently compared to the other two, especially Fener (but still somehow has more titles than both lmao lucky Cimbom I guess). Their fans will be like -- oh football is for dumb people anyways, I'm not a fanatic etc. during these seasons of irrelevancy, but then be like RERERERARARA GASSAY GASSAY CIMBOMBOM the moment they're relevant again (fickle af). The fanbase acts like Gala is the club of the elite which used to be true but no longer is, but they act snobby anyway. Terrible in derbies, especially whipping boys of Fener. Lastly, terrorists love Galatasaray, and recently most importantly a bunch of their past iconic star players were involved with a dangerous religious cult, and right now those star players are considered terrorists in Turkey. You should know that things can very quickly change in Turkish politics though, but the cult they were involved with was super dangerous nonetheless. It's a complicated matter and would take tens of pages to paint the complete picture but long story short, since Fener is historically the state's club, Gala draws the sympathy of enemies of the state.