There are generally better restaurant options elsewhere around the city, but if you are dead-set against crossing any of the rivers there are a number of good options on the North Shore.
My favorite of all Pittsburgh restaurants on the North Shore is Max's Allegheny Tavern. They serve very traditional German food in a quaint, casual atmosphere. I've had most everything on their menu and end up eating there at least once every two weeks or so. They have very good sausage and veal plates. I really like their sauerbraten as well as their goulash. They have perhaps the best Reuben in the city. For sides the spatzle and fresh-cut fries are top notch.
Behind that I would mention Hyde Park. It is a steakhouse and part of a chain, but they serve top notch steak and sides. The atmosphere and prices are commensurate with Pittsburgh fine dining restaurants.
After that, I would send you to Legends of the North Shore. It is typical red sauce Italian, but everything is very good. They have a tiny dining room, and it's BYOB. Very intimate.
Bettis' Grille 36 is also very good. It is a sports bar/restaurant, but the activity in the large bar rarely affects the dining rooms. It is part of a chain, but they serve very good food at reasonable prices and have a few tasty twists on common pub food. Casual and solid.
Next, I would suggest Rivertowne North Shore. It is typical bar food. Lots of sandwiches and fried things, all of which are super tasty. They also have a large beer selection. It is very good, very casual and not very expensive.
After that you're pretty much in the weeds on the North Shore, I think. Atria's, a chain with a joint in PNC Park is good, casual dining. Nickey's Thai Kitchen, on Western Avenue, is pretty good if you're looking for some Thai food. If you're just looking for sandwiches or something then there is a Peppi's as well as the Allegheny Sandwich Shop, both also on Western.
Max's Allegheny Tavern
537 Suismon St, Pittsburgh, PA 15212