My government has an information page specifically about what you should do when this happens. Basically: register the birth locally (which gets you the local papers), register at home as soon as possible, and you may want to convert the birth certificate for ease of paperwork. There’s another page where you can look up the specifics for each country, but most of them (including Turkey) don’t have any specifics.
This registering abroad can cause issues when giving birth in countries like the USA, where being born within the border makes you a citizen, and therefore liable to pay tax even if you have a dual nationality and live abroad. This is especially weird because the Netherlands does not normally permit dual citizenship. Sounds like a fun bit of paperwork!
Your best bet, for any country, will probably be “contact the embassy and ask what you need to do”. Or maybe “don’t travel across the border when you’re about to give birth”; I can’t imagine a nine month pregnant woman will have a great time on vacation anyway.