Tower Local Control trainee reported a departing aircraft reported flight control problems and returned to the airport.

Narrative

During this incident I was training on Local Control; the situation began when I departed Aircraft X off Runway 27R. I stated to Aircraft X 'fly straight out Runway 27R cleared for take off'. The pilot of Aircraft X repeated said instructions and began their takeoff roll down runway 27R. From the beginning of take off roll to the lift of Aircraft X there appeared to be no issues regarding the aircraft. Aircraft X started their right cross wind unexpectedly while they appeared to be at 1500 ft. according to the radar scope. The pilot of Aircraft X stated ' Pontiac Tower I need to Re-land; I have some unexpected shimmying'. In response instructed Aircraft X to start flying eastbound and cleared Aircraft X to land on Runway 27R and stated 'Do you require any assistance'. The pilot stated 'Currently no; but I have some shimmying here'. Amongst this situation myself and my trainer instructed other aircraft inbound into the field to extend their downwind to ensure the safety and control of Aircraft X. Once further in the downwind Aircraft X appeared to be starting their base turn for RWY 27R. The pilot was asked if they preferred to land on Runway 27L; and the pilot of Aircraft X began their right base turn and stated ' I'm in an uncontrolled turn right now; Hold on'. Aircraft X then keyed up again and [requested priority handling].Aircraft X's base turn appeared to be too sharp and they turned too far out of being able to Land on Runway 27R. My instructor keyed up and told Aircraft X 'Change to Runway 27L; Runway 27L cleared to land' Aircraft X then tried to crank their Right base turn in to land on Runway 27L. After leveling out the turn Aircraft X Landed on Runway 27L safely without any assistance. Aircraft X was then instructed to turn off of the runway and to contact Ground Control.