The explanation of shear rate in laminar flow is straightforward: We imagine small layers of fluid that glide on each other. Now, in turbulent flow, this does not work as there are no layers. If I want to know the apparent viscosity of a shear thinning (or other non-Newtonian) liquid, I need to know the shear rate. I've asked about this before here, and received an answer. However, I can't solve the Navier Stokes equation, so someone has to walk me through it or present me with an answer.
Fluid may be assumed to be a power-law fluid.