Entering Türkiye from an IST layover

Visa-free is common and specific

A large set of European passports, the UK, and the US can enter for short tourism stays, often up to 90 days in any 180-day period. The exact allowance is on the Turkish MFA information for your nationality, not in a group chat.

If your passport is not visa-free, get the e-Visa before you fly to the connection, not after you land. The official site is evisa.gov.tr. Other domains that look similar are not the government.

Passport and onward flight

Travel with passport validity that satisfies the current rule for your nationality, the onward boarding pass, and the hotel address if you have one.

A refused boarding on the inbound, because the airline does not like your entry papers, happens before you ever see Istanbul. Check with the carrier.

Mosques are not a visa issue, but they have rules

Dress modestly for Hagia Sophia and the Blue Mosque. Carry a scarf if you will want one. This has nothing to do with immigration and everything to do with whether the 24 hour plan works when you arrive.

Staying airside

If the e-Visa is not going to arrive in time, or you do not qualify, stay in transit. IST is built for long connections. The old city is not going to be good if you spend the layover solving a visa.