I have 2 separate tickets Singapore — Istanbul Atatürk (IST) and IST — Lviv. Is it possible to claim checked-in luggage and check it in on the next flight, without going through passport control forth and back in Ataturk airport?
No it is not possible.
Baggage collection in Istanbul Ataturk airport occurs after (inbound) immigration, and baggage drop occurs before (outbound) immigration.
Depending on the airlines involved, the original airline may be able to check the bags all the way through to the final destination which would remove the need to pass through immigration. If this is not possible the you will need to clear immigration in order to claim and re-check your bags.