Suppose Alice Jones nee Smith got married in Turkey, and she is issued a Turkish marriage certificate which certifies her new married name, and then she moves to the US and applies for a driver license which requires her birth certificate and name change document in her state. Is her Turkish name change document just as valid as an American issued one would be?
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.
Sometimes, an agency or institution will also reserve the right to not honor a document if the relevant civil servant has a reasonable belief based upon some relevant fact, that the document is a fake.
But, some agencies or institutions could have more relaxed, or more cumbersome, requirements by adopting regulations that say so.
For example, many states have laws that allow you to do a name change for purposes of a driver's license from a pre-marital name without any other proof other than simply by declaring under oath that you got married on a particular day and changed your name, since that situation is rarely suspicious and covers the vast majority of cases.
An alternative option would be to bring a name change petition to a U.S. court and to have it declare the the Turkish name change is valid, usually with essentially the same kind of evidence, but sometimes with the court allowing you to explain why you can't comply with some formality, because, for example, your country is at war and isn't issuing apostilles or your country doesn't have the concept of notarizing or certifying this particular kind of document, or because you would face persecution in that country if you had to go back to it in order to get that paperwork.
Once this was done, all U.S. government, state, and local agencies and institutions would have to obey that court order, and if there are lots of agencies or institutions that have different or onerous requirements, this may be easier to do.
There are also some states that recognize the concept of a common law name change. In these states, any name (and sometimes all names) that you voluntarily go by, or answer to, are legally your name, even if you don't have any paperwork to back it up.