"Really" has two means in English, often depending on its position in the sentence: "in actuality" and "very/a lot."
For example: He really is handsome. = (Actually, he is handsome.) He is really handsome. = (He is very handsome.)
I think that "Istanbul has really changed" should be accepted because it means the same thing as "Istanbul has changed a lot." It's NOT the same thing as "Istanbul really has changed," which I believe should indeed be translated with "gerçekten."
Thanks for the response! But I'm afraid I have to disagree. The Oxford English Dictionary also doesn't say anything about "really" being more intense than "very." Instead, it says that, "really" is synonymous with "very; thoroughly." As a native speaker and ESL teacher of 9 years, I concur. The only difference I perceive between "really" and "very" is that "very" feels to me more stiff/formal. Thus, I feel that "has really changed" should be accepted. :)