I read that for a turbulent boundary layer, a thicker (turbulent) boundary layer separates earlier than a thinner (turbulent) boundary layer? Why is that? Or is it the other way around?
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.
Turbulent boundary layer is thicker than laminar boundary layer as rightly pointed out, for any given distance travelled by fluid over a surface. However, thickness is not directly related to separation. The separation results when the fluid doesn't have enough energy (read momentum) to counter an adverse pressure gradient (typically a divergent section). Turbulent flow has more mixing between the layers and can survive more adverse pressure gradient than a laminar flow (which has no mixing between the layers). So for the same surface geometry, a turbulent boundary layer would separate later (or may not even separate) compared to a laminar boundary layer. The same idea is used in turbulators or vortex generators to keep the flow attached to the surface.
Not a textbook explanation, but maybe clarifying anyway.
Laminar flow or a jet doesn't stop unless something makes it stop. They are comparatively homogeneous throughout the layer, because they have no internal conflict. The whole thing wants to be a layer, because forming a layer is the path of least resistance. The same doesn't go for turbulent layers. They are forced to be layers, so they tend to generate a difference between the 'surfaces' and the 'center' of the layer, which gets more influential, the thicker the layer gets. Turbulence does not want to be a layer, it doesn't have any reason to. It wants to be a bubble. So as soon as there is enough turbulent mass gathered in a layer to do that, it will, which results in separation.