We consider the flow of quasi-Newtonian fluid through a thin porous media. The media under consideration is a bounded perforated three dimensional domain confined between two parallel plates, where the distance between the plates is described by a small parameter ε. The perforation consists in an array of solid cylinders, which connect the plates in perpendicular direction, with diameter of size ε and distributed periodically with period ε. The flow is described by the three dimensional incompressible stationary Stokes system with a non-linear viscosity following the Carreau law. We study the limit when the thickness tends to zero and prove that the averaged velocity satisfies a non-linear two dimensional homogenized law of Carreau type. We illustrate our homogenization result by numerical simulations showing the influence of the Carreau law on the behaviour of the limit system, in the case where the flow is driven by a constant pressure gradient and for different geometries of perforations.