We investigate the role of the coupling between a spin-orbit coupled semiconductor nanowire and a conventional s-wave superconductor on the emergence of the topological superconducting phase with Majorana bound states in an applied magnetic field. We show that when the coupling is strong, the topological phase transition point is very sensitive to the size of the superconductor and in order to reach the topological phase a strong magnetic field is required, which can easily be detrimental to superconductivity. Moreover, the induced energy gap separating the Majorana bound states and other quasiparticles in the topological phase is substantially suppressed compared to the gap at zero field. In contrast, in the weak-coupling regime, we find that the situation is essentially the opposite, with the topological phase emerging at much lower magnetic fields and a sizable induced energy gap in the topological phase, which can also be controlled by the chemical potential of the superconductor. Furthermore, we show that the weak-coupling regime does not generally allow for the formation of topologically trivial zero-energy states at the wire end points, in stark contrast to the strong-coupling regime, where such states are found for a wide range of parameters. Our results thus put forward the weak-coupling regime as a promising route to mitigate the most unwanted problems present in nanowires for realizing topological superconductivity and Majorana bound states.