"I am going to university in Istanbul."
Translation:İstanbul'da üniversiteye gidiyorum.
Your sentence is ok but it is not a translation of the English sentence given here. It means "I am going to the university in İstanbul."
In the sentence given, "İstanbul'da" is a phrase describing the location where the whole sentence takes place. ("I am going to university." "WHERE?" "In İstanbul")
In your sentence, İstanbul'daki is similar to an adjective modifying "university." It distinguishes the university you are talking about from other universities. ("I am going to university." "WHICH?" "The one in İstanbul.")
Native Turkish speakers please correct me if I'm wrong, but I think the dealio is that "Istanbul'da" is an adverb, it describes "gidiyorum" and "Istanbul'daki" is an adjective, it describes "üniversiteye". I also find it confusing because I don't think theres a rule in English for when a prepositional phrase is more appropriate than a prepositional clause.