The Verdict: Not eligible
Turkey does not recognise same-sex marriage, so the marriage gives no route to citizenship or a family residence permit.
Article 134 of the Turkish Civil Code provides for marriage only between a man and a woman, and a same-sex marriage concluded abroad is not registered.
You should also know that draft criminal provisions targeting LGBTI people have been proposed in Turkey's recent judicial reform packages. We have not confirmed whether any have been enacted. Take independent advice before relying on the relationship in dealings with the Turkish authorities.
Other routes do not depend on your relationship: residence through property or investment, work, and naturalisation after five years. Our other Turkey quizzes cover them.