I’d be nervous too, but the agent wasn’t totally wrong. TK’s system usually won’t sell an impossible connection, especially at their own hub. Istanbul is built for short transfers.
Coming from the US, you normally do not reclear security for international to international connections at IST. You just follow the transfer signs and go straight to the next gate area. That’s a huge difference compared to the US.
Business class helps since you’ll be off the plane quicker, and gates are often closer together for long haul banks. The real risk is delays out of JFK. If you land on time, an hour is usually fine. If you’re late, it gets stressful fast.
I wouldn’t relax, but I wouldn’t panic yet either. Worst case, TK is pretty good about rebooking if it’s their delay.