I would say no, stabilizar bar wouldn't be sole cause. I would look at the steering components first. Tie rods, pitman arm, idler arm, steering box. Then ball joints and control arms. Bad alignment can also cause it. It may be a combination of worn parts. You can inspect your steering and suspension for wear with a pry bar, looking for loose (worn) connections, also have someone move the steering wheel back and forth while you observe each connection. It could be a loose nut, check all to make sure they're properly torqued. Steering and ball joints fit into tapered holes. Besides the joint itself, look at the holes they fit to make sure they're still tight and not enlarged. Jack up the lower control arm and put a bar under the tire, lift up and down to see if the lower ball joints are loose. You can get loaner tools with just a refundable deposit to change out all these items yourself if you're on a budget.
Edit: also look for cracks, bent or broken components.
