I am trying to shut off the gas to my stove so I can replace it. It turned clockwise and the pilot lights went out, but it just continues to turn several rotations. Is this normal? If so, how do I know it is off?
There is a small knob/handle on one side, and on the other side is a nut/end of bolt. When I turn the handle, that nut turns as well.

Stop turning when the handle is perpendicular to the pipe and valve. I would consider replaceing that valve with a new one that stops after a quarter turn.

There is nothing wrong with the valve. Changes have been made to the codes because of the older valves. Operation, like you have seen and the packing material as a source of leaks.

Your likely grandfathered for now, but consider changing it with a new installation.