Assuming that you are flying domestic -> international, Turkish Airlines would through check baggage on separate tickets to BA upon request so yes at both airports, you could avoid the chaos of picking your bags, rechecking them, etc...

Assuming that you have a boarding pass, at IST follow signs for international connections after you arrive and that would lead you to a dedicated passport control and transfer security checkpoint for domestic -> international flights. At SAW, there is also a dedicated international transfers path before arriving at the baggage claim hall but that is much more easier to miss. However, the landside path at SAW is much shorter than IST.

If you don't have a boarding pass at both airports, you would be not allowed to use the shortcuts and BA doesn't have a transfer desk at both airports regarding connecting passengers from domestic to international so if you don't have a boarding pass, even if your bags are checked through, you would need to go out at wait for the desks to open to receive your boarding pass.