We consider the flow of a generalized Newtonian fluid through a thin porous medium of thickness ε, perforated by periodically distributed solid cylinders of size ε. We assume that the fluid is described by the 3D incompressible Stokes system, with a non-linear viscosity following the Carreau law of flow index 1 < r < +∞, and scaled by a factor εγ , where γ ∈ R. Generalizing (Anguiano et al., Q. J. Mech. Math., 75(1), 2022, 1-27), where the particular case r < 2 and γ = 1 was addressed, we perform a new and complete study on the asymptotic behaviour of the fluid as ε goes to zero. Depending on γ and the flow index r, using homogenization techniques, we derive and rigorously justify different effective linear and non-linear lower-dimensional Darcy’s laws. Finally, using a finite element method, we study numerically the influence of the rheological parameters of the fluid and of the shape of the solid obstacles on the behaviour of the effective systems.