In general, you cannot apply for a Chinese visa from a country of which you are not a citizen or permanent resident.
Specifically for Turkey (my emphasis):
(3) Proof of legal stay or residence status(applicable to those not
applying for the visa in their country of citizenship)
If you are not applying for the visa in the country of your
citizenship, you must provide the original and photocopy of your valid
certificates or visa of stay, residence, employment or student status,
or other valid certificates of legal staying provided by the relevant
authorities of the country where you are currently staying.
You'll note visitor is not a valid status. This means that you'll have to apply in advance for your visa from the US.
Although the link in my answer is to
chineseembassy.org, apparently this is the official domain for foreign missions as per the Ministry of Foreign Affairs' website.