The effect of weak shear Thickening and shear thinning on the stability of the Taylor-Couette flow is explored for a Carreau-Bird fluid in the narrow-gap limit. The Galerkin projection method is used to derive a low-order dynamical system from the conservation of mass and momentum equations. In comparison with the Newtonian system, the present equations include additional nonlinear coupling in the velocity components through the viscosity. Similar to Newtonian fluids, there is an exchange of stability between the Couette and Taylor vortex flows. The results indicate that with increasing shear-thickening effect, the flow turns into unsteady state at higher Taylor numbers. In contrast, with increasing the shear-thinning effect the flow turns into unsteady state at lower Taylor numbers. Moreover, the results are in accordance to the other present research results.