Hydrodynamic lubrication is studied for both shear thinning and viscoelastic polymer solutions. We find that elasticity, notably strong normal stresses, does not change the friction significantly for the range of parameters tested in this manuscript. Shear-thinning properties, on the other hand, do change the formation of the lubricating layer thickness and the dependence of friction on velocity relative to Newtonian fluids. A hydrodynamic model that includes shear thinning is developed and compared to experimental data. The model describes the dependence on lubrication parameters well, but underestimates the lubricating layer thickness by a constant factor of roughly 1.5. The theory allows us to define a Hersey-like number for shear-thinning fluids that describes the lubricating layer thickness as a result of the balance between normal load and viscous force. For each tested liquid it succeeds in collapsing friction measurements onto the same curve. The friction analysis for both lubrication theory and experiments then reveals that shear thinning mainly changes the layer thickness, which is the main determinant of the friction coefficient.