Isothermal transformations are minimally dissipative but slow processes, as the system needs to remain close to thermal equilibrium along the protocol. Here, we show that smoothly modifying the system-bath interaction can significantly speed up such transformations. In particular, we construct protocols where the overall dissipation W diss decays with the total time τtot of the protocol as, where each value α > 0 can be obtained by a suitable modification of the interaction, whereas α = 0 corresponds to a standard isothermal process where the system-bath interaction remains constant. Considering heat engines based on such speed-ups, we show that the corresponding efficiency at maximum power interpolates between the Curzon-Ahlborn efficiency for α = 0 and the Carnot efficiency for α → ∞. We confirm our analytical results with two numerical examples where α = 1/2, namely the time-dependent Caldeira-Leggett and resonant-level models, with strong system-environment correlations taken fully into account.