In this study, we report on the analysis of the magnetic domain wall (DW) curvature due to magnetic field induced motion in Ta/CoFeB/MgO and Pt/Co/Pt wires with perpendicular magnetic anisotropy. In wires of 20 μm and 25 μm, a large edge pinning potential produces the anchoring of the DW ends to the wire edges, which is evidenced as a significant curvature of the DW front as it propagates. As the driving magnetic field is increased, the curvature reduces as a result of the system moving away from the creep regime of DW motion, which implies a weaker dependence of the DW dynamics on the interaction between the DW and the wire edge defects. A simple model is derived to describe the dependence of the DW curvature on the driving magnetic field and allows us to extract the parameter σE, which accounts for the strength of the edge pinning potential. The model describes well the systems with both weak and strong bulk pinning potentials like Ta/CoFeB/MgO and Pt/Co/Pt, respectively. This provides a means to quantify the effect of edge pinning induced DW curvature on magnetic DW dynamics.