In this study, the two-dimensional creeping flow of shear-thinning fluids through a square-edged concentric orifice in a pipe has been analysed numerically over a wide range of orifice plate thickness, 1/16 ≤ γ ≤ 1 with fixed diameter ratio of the orifice to pipe, β = 0.5. The numerical result has been derived over the range of power-law index, 0.2 ≤ n ≤ 1 to analyse the discharge coefficient as an important parameter while using an orifice meter as a flow measuring device for shear-thinning fluids. The influence of the orifice plate thickness (γ) and power-law index (n) on the discharge coefficient has been studied in detail. Further flow field has been examined in terms of velocity contours, viscosity contours, and streamline patterns over the range of thickness of the orifice and power-law index. A comparison has been made between two different types of pressure tapings, i.e., D−D/2 taps and corner taps for the entire thickness-to-diameter ratio taken into consideration. In addition, a correlation has been established for the discharge coefficient as relation to orifice thickness and power-law index thereby enabling their interpolation for the intermediate values in various applications.