Our Best of Istanbul tour stayed at the Haze Hotel. It has a wonderful restaurant on the top floor with a case of dishes from which you can choose. I think they had some bar seating.
ETA: Mükellef Karakoy, reservations recommended on weekends at least
(You can see the Galata Tower from the street in front of the Haze.)
If you're looking for something friendly and very casual, there is an alley near the Haze with loads of small cafes and restaurants and lots of outdoor tables. You'll know you've found it by the colorful overhead umbrellas... we called it Umbrella Alley, but it's Mumhane Cd.
We did a progressive Turkish street food style dinner there, starting at the end near the Haze with stuffed mussels and working our way down the street, having soup (choice of 16 kinds) and eventually dessert.
ETA: Gala KARAKÖY (not a restaurant, just a little place to get started with stuffed mussels)
Around the corner is a friendly little bar - with 3 or 4 outdoor tables and a super friendly bartender, maybe the owner. He let us bring our mussels paired with glasses of his Turkish wine. Across from the bar is a small place where you can finish your progressive dinner with the Turkish dessert of chocolate and fruit. If I can find the names, I'll add them.
ETA: REDS Coffee & Chocolate (can't find this on Google Maps)
I can't find anything with the name of the friendly bar, but it's directly across the alley from REDS
None of this is fancy, it's all fun and friendly. And if you can't make friends with that bartender, well.... at least you'll have a fun meal!