Many Turks are also against Erdogan, and Istanbul has been a center of opposition, so I don’t see a problem with visiting there. But you could donate to one of the opposition parties or candidates, or to a Kurdish organization (but not until after you leave Turkey).
Yeah, I suppose a little bit of “indirect support” is inevitable given that many of the tourist sites are government-run, but otherwise, I don’t think just visiting Istanbul and spending money there constitutes support for the Erdogan regime any more than visiting New York on vacation supports Trump. In both cases, if you’re doing most of your spending at locally owned businesses, much of it is likely going to people who don’t support the current government.
(That said, I’m not sure I would apply this argument to, say, El Salvador, where “making the country safe so it can be a tourist destination and bring in the US dollars” is a core part of the regime’s justification for its actions. It’s a spectrum.)
It’s my understanding that tourism to the US has dropped significantly because people don’t want to support Trump in any way (and/or are afraid about how they may be treated by the government while here).
I spent six days in Turkey last Thanksgiving. (Aside: how many people can say they went to Turkey for Thanksgiving? 
)
We spent three nights in Istanbul and three nights in Cappadocia.
Istanbul was incredible. I highly recommend it. Highlights that I enjoyed were visiting the Hagia Sofia (and nearby museum), the Blue Mosque, the Basilica Cistern (which is unlike anything you can likely imagine), Topkapı Palace, the Grand Bazaar, and finally, spending an afternoon in a 300-year old hammam for a Turkish bath experience. I would go back again in a heartbeat just for the hammam.
If you stay overnight, I highly recommend getting a hotel in the old town on the peninsula near Sultanahmet Square. Everything I’ve mentioned above is within easy walking distance of each other. Traffic is otherwise a nightmare…like trying to drive around Manhattan. It took over an hour to get to the airport, and most of that time was spent trying to get out of the city.
Apparently true. It’s been a few years since I was there, and of course I adopted the careful visitor protocol of not raising politics in conversation. But when the subject came up obliquely, that sentiment was expressed.
I guess Turkey is a deeply divided country between Islamic fundemantalism and secular modernism. It will be interesting to see how this eventually plays out….