How does the turbulent boundary layer thickness affect separation?
On a flat plate the thickness of the boundary layer grows along the flow path without a tendency to separate. Therefore, thickness alone is not the culprit.
However, when the pressure gradient in flow direction becomes positive (increasing pressure), the flow will slow down and thickness grows much faster than in non-decelerating flow. Now separation happens when the speed at the surface drops to zero. Right before separation you will see an exponential growth of the boundary layer, but it is only an indication for the decelerating flow which really is to blame for the separation.