Where the passport was issued makes little or no difference. I've been in and out of Turkey several times on a passport issued by the British Consulate in Istanbul with an FCO mark on it.
If you're leaving the country after changing passports you just need to show both of them to prove that you had a valid visa. Don't worry though if you aren't able to show the old one because they'll be able to check all your previous entries and exits on their computer system.
This is the old one that I showed them, thanks to Molly, when I changed over. (Another good reason to keep photocopies of important documents)
