I'm traveling on Turkish Airlines from Tbilisi, Georgia to Frankfurt, Germany, with a connection at Istanbul Atatürk International Airport. I arrive in Istanbul at 6:50 am and my next flight to Frankfurt at 7:55 am. Is 1 hour and 5 minutes enough time to make the connection?
Generally, you should be able to make a connection in just over an hour at Atatürk, particularly at such an early time in the morning. As a Turkish Airlines transfer-transit passenger, you simply disembark when you arrive, with only your cabin baggage, and go directly to the boarding gate without any flight or passport procedures. If you haven’t gotten your boarding pass for the connection, go to the Turkish Airlines Transit Check-in desk.
You don’t say whether you have a single booking, but you do mention just Turkish Airlines as your carrier. If you miss your connection, the airlines should assist you, placing you on a later flight, and usually without charge. However, that may depend on the conditions under which the connection was missed and how you're ticketed (single or separately).
Connecting in Istanbul Airport can be incredibly variable depending on the exact situation.
Some flights will arrive at the terminal itself, which will mean a walk of between a few minutes and maybe 5-10 minutes depending on which gate you arrive at and which of the security facilities are open.
Other flights will arrive at "bus gates" which will mean having to catch a bus to the terminal. Depending on which of the bus gates you park at, and which bus you end up getting on (as there will be multiple per flight), it can take 20 minutes or more to make it to the terminal from when the plane door opens.
Once you're in the terminal there are two potential paths you can be forced to take.
If you arrived from a "clean" airport (from a security perspective), then you will generally be allowed to go straight into the departures area. I say generally, because sometimes they will forget to send a staff member to open the connection into that area. And sometimes even if there is someone there they will not allow you to take this route.
I suspect that Tbilisi is not considered a "clean" airport, so this option probably won't be available to you.
If you can't proceed directly to the departure area, then you will need to go through transit security. This can range from a few minutes to well over 30 minutes depending on the time of day, how many flights landed before you, and how many security areas they have open.
Then you can go to your gate, which could take another 5-10 minutes depending on which gate you're departing from. And again, this could be a bus gate.
If your inbound flight is on time, and if you don't get a bus gate, and if the line at security isn't long, then 65 minutes is tight but doable. In many other situations, it will not be.
I travelled through Ataturk recently, connecting from TK to TK, my family was travelling through Ataturk at about the same time, also connecting from TK to TK, however different flights to mine.
I arrived ahead of them and noticed their inbound flight was delayed by 30 minutes.
I asked what is their process for passengers whose inbound delay is putting the connections at risk. His response was - there is no process!
I rushed to the transfer desk located somewhere in the middle of the terminal or around gate 215. There I asked which gate my family's inbound was arriving at. I was told gate 219, and that flight arrival and departure levels are the same, making transit much more seamless.
I already knew their connecting flight was leaving from gate 226, so I met them and gate 219 and rushed them across to gate 226. By having done what I did, I saved them time looking for onward flight details, which would have lost them at least 5-10 minutes of precious time remaining for connection.
They made the connection and boarded the flight when on final call thanks to my involvement when I would have expected from Turkish ground staff to have either met them off the inbound flight and assisted with tight transit, or at least radioed to cabin crew their onward flight gate to help with timely connection.