"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. :)
'He's really very handsome' in English would naturally be taken as countering the idea that he's not really handsome. It's not a neutral statement. Sure, in context, and with the right intonation, the two could have the same meaning but it's a question of what's natural in English, surely, as a translation of the Turkish phrase.
Surely the real question here is, has the learner correctly understood the meaning of the Turkish phrase. I can't imagine a situation in which an English speaker learning Turkish would say 'Istanbul has really changed' if they had not correctly understood 'çok değişti'. You can argue the toss about fine distinctions of meaning within the range covered in English by 'really', but that is not the point for learners who simply want to have their comprehension acknowledged and move on to the next question. It starts to feel like marking 'Istanbul has really changed' as wrong is penalising learners for not agreeing with some people's slightly idiosyncratic understanding of 'really'.