Hi Natalie,
What a fabulous trip you have planned! Turkey is such a cornucopia of treasures. A year ago, I did a 3 week solo trip bumping around the country with no pre planned itinerary on a similar route as yours.
May I offer a suggestion? Fly from Istanbul to Capadoccia and Capadoccia to Antalya. Pick up your rental car in Antalya and drop it off in Izmir or Deniliz and fly back to Istanbul. Hear me out. Istanbul has some of the worst, nail biting traffic I’ve ever seen. I’d hate to navigate it as a driver. I love a good road trip but the scenery to and from Capadoccia is pretty desolate.. Flights from IST to either Kayseri or Nevsehir (Capadoccia) on Turkish Airlines or Pegasus Air run about $25 one way. I bought my tickets the day before flying on both these legs and it was under 30 with an upgrade. Ditto from there to Antalya. This will save you frustration, a couple days of driving and a hotel room or two enroute.
Your hotel in Capadoccia can arrange cheap airport transfers, ground tours and or balloon rides while there. I stayed at the Stone House Cave Hotel in Goreme. They arranged all three for me for less than I could arrange on my own. I would recommend them as well as Pumpkin restaurant in Goreme. From there, I flew to Antalya on an evening flight, giving me an extra full day in Goreme.
I really like Antalya, extending my time there a few days. When traveling in Turkey, I found the app Moovit much better than google maps or Citymapper. Antalya has great public transit and taxis are cheap. Pick up your rental when you leave.
Your itinerary leaves off Pammukkale outside Denizli. I’d take it well before Canakkale. You could hit it after Bodrum before Alicati in route then fly back to Istanbul from Izmir. Or do it after Alicati and fly home from Denizli. (Warning Denizli airport is waaaay out of town in the middle of nowhere.)
Have you looked at flying open jaws into Istanbul and back home from Izmir or Denizli? That would save you another travel day. Feel free to shoot me any questions you might have. I hope you have a fabulous trip!