By considering the entrainment effect and the wall effect, the present study has been made to extend the applicability of a high Reynolds $k-\varepsilon-\gamma$ model to wall bounded shear flows. The extension of the model was realized by introducing variable coefficients for the near wall, low Reynolds corrections in the turbulent dissipation rate equation and the eddy viscosity. The proposed model by the present study was tested against various wall bounded shear flows such as turbulent pipe flows, channel flows, flat plate boundary layer flows, wall jet flows and transitional boundary layer flows. The results show that the proposed model correctly predicted the Reynolds shear stress and the turbulent kinetic energy near the wall. Agreements with the experiments of pipe, channel and flat plate boundary layer flows were generally satisfactory. Comparison with the experiments of wall jet flows and transitional boundary layer flows revealed that more study is needed to correctly simulate those flows.