Each state, each local government, and each government agency can make their own rules on this question (if it was another U.S. state, a provision called the "full faith and credit clause" requires all other states to honor it).
Usually, the Turkish name change document will be honored by any U.S, state or local government agency or institution if (1) the name change document or a copy of it, is certified or notarized, (2) there is an apostille from the appropriate Turkish national government official stating that the certification or notarization is valid, and (3) there is a translation of the document into English that accompanies the original document or certified copy of the document that is supported by a sworn certification from a foreign language translator that the translator is qualified to translate the document and that the translation of the document is true and correct.
But, some agencies or institutions could have more relaxed, or more cumbersome, requirements by adopting regulations that say so.