Yes. To me it sounds like something is to loose and goes into oscillation while cornering.

I'd look at all the linkages in the front suspension including the steering links. Check for looseness and or breakage. Also look the ball joints. You will need to jack the car up and possibly remove the tire for some of these checks. When you try to wiggle things, put some muscle into it. After all the car's whole weight is on these components.

Not knowing the specific car's design, I'd bet it has a trailing arm rear suspension. I'd check the linkages used to keep the arms straight, and the condition of the arm it's self. I had one on a Chevy Nova go bad so the arm flexed more than it should. As such it effected the handling.