Upvote:1
An incomplete but perhaps useful answer: on United's page concerning in-cabin pets, they note that
United's in-cabin pet policy applies to flights operated by United and United Expressยฎ only. For codeshare flights, please check the operating carrierโs in-cabin pet policy. If you are traveling on multiple airlines, please check with each carrier individually to ensure acceptance. United can only accept your pet for flights operated by United and United Express.
(bolding mine)
There doesn't seem to be similar language on their pages concerning pets as cargo, but I wouldn't necessarily take that to mean the policy is different. It's also possible that Turkish Airlines will guarantee that your pet will be able to travel on all code-share segments if you buy through them, even though United won't; but that seems doubtful to me.