The generation of large spin currents, and the associated spin torques, which are at the heart of modern spintronics, has long been achieved by charge-to-spin conversion mechanisms, i.e., the spin Hall effect and/or the Rashba–Edelstein effect, intrinsically linked to strong spin–orbit coupling. Recently, a novel path has been predicted and observed for achieving significant current-induced torques originating from light elements, hence possessing weak spin–orbit interaction. These findings point out to the potential involvement of the orbital counterpart of electrons, namely the orbital Hall and orbital Rashba–Edelstein effects. In this study, we aim at quantifying these orbital-related contributions to the effective torques acting on a thin Co layer in different systems. First, we demonstrate in Pt|Co|Cu|AlOx stacking a comparable torque strength coming from the conversion due to the orbital Rashba–Edelstein effect at the Cu|AlOx interface and the one from the effective spin Hall effect in the bottom Pt|Co system. Second, in order to amplify the orbital-to-spin conversion, we investigate the impact of an intermediate Pt layer in Co|Pt|Cu|CuOx. From the Pt thickness dependence of the effective torques determined by harmonic Hall measurements complemented by spin Hall magneto-resistance and THz spectroscopy experiments, we demonstrate that a large orbital Rashba–Edelstein effect is present at the Cu|CuOx interface, leading to a twofold enhancement of the net torques on Co for the optimal Pt thickness. Our findings not only demonstrate the crucial role that orbital currents can play in low-dimensional systems with weak spin–orbit coupling but also reveal that they enable more energy efficient manipulation of magnetization in spintronic devices.