Understanding the Viscous Sublayer in Turbulent Flow

I’m slightly confused at how/why the viscous sublayer forms in a turbulent boundary layer. From what I understand, the boundary layer is a region where there is momentum transport through the planes parallel to the flow.

In the laminar region, the momentum transport is purely a result of viscous diffusion/molecule attraction.

In the turbulent region, the momentum transport is a result of viscous diffusion + bulk transport caused by turbulent eddies.

Since the momentum transfer caused by eddies is significantly larger than the one caused by viscous diffusion, turbulent flow is dominated by turbulent shear stresses, except at the viscous sublayer. Why is that? Is it because turbulent shear stresses are effectively zero at the boundary due to the cross-correlation term, $bar{u^{prime }v^{prime }}$, being zero? If that is the case, then why does the viscous sublayer get smaller as Re increases?