Key sights of Istanbul in 3 hours?

Contact SRM Travel. They do the Turkey and Istanbul tours for Rick Steves. Just go online or look them up in Rick's Istanbul guide book--the owners wrote the book. You can probably arrange for a 1/2 day tour. Have Fun!

With that limited amount of time I think you might be better off just getting a driver or cab to take you into Sultanahmet. From there you can do a bungee visit of some of the top sites in the city. If you decide to go that route, here are some ideas along with the time needed to conduct a blitz visit. All are right next to each other, but you'll need to allow a few minutes of travel time between them. My recommendations, in order, based on the amount of time you have and the interests you listed are:

1. Aya Sophia: 1 hour - This is my top site on the list. I recommend that you see if first so you don't miss it .
2. Blue Mosque: across the street from the Aya Sophia, 20 to 30 minutes. Pay attention to the time of day. It will be closed for about 30 minutes during the five daily prayer times.
3. Arasta bazaar: a small bazaar next to the Blue Mosque. It's nothing like the Grand Bazaar, but will give you a flavor. 20 minutes to walk through, longer if you want to shop
4. Topkapi Palace: 2 hours - You would be hard pressed to see this site alone in 3 to 4 hours. I typically recommend 6 hours. You'll have to skip the harem, walk fast and hit only a couple of the high points (e.g. the treasury).
5. Hippodrome: 20 to 30 minutes

Good luck!

Hagia Sophia ( an oh my moment) and the Grand Bazaar would provide a good 3 hour sojourn.

It's probably best to focus on enjoying a couple of sights rather than trying to cram it all in.

The Grand Bazaar is a must, and too much time there can be over-whelming, so it's a good place to go for a couple of hours.

But Sultanahmet is a good place to head for. The Blue Mosque is free and gorgeous to look around and photograph. And right there is the amazing Hagia Sophia - it's a short stroll between the two, and they're both memorable.

One of my favourite spots was the Archaeology Museum, which is right next to the Topkapi Palace. It's amazing. Their collection of historical artifacts is very impressive. I'd opt for the Museum over the Palace - especially since you only have a short time slot.

In that area are quite a few small local lokantas for a cheap lunch. Make sure you take time to sit and drink in the sights and sounds of the city.

Go to Sultanahmet Square and do not spend any minutes indoor places. Check them from outside except Grand Bazaar.

-Blue Mosque
-Hagia Sophia
-Grand Bazaar (30 mins)

Take the T1 and get off at Karaköy. Take the Tünel.

You will find yourself in the center of local life. It's Istiklal Avenue and the heart of Istanbul. Walk toward Taksim Square.

NOTE: If you haven't been to Istiklal Avenue, you haven't been to Istanbul.

Good luck.