The Magellanic Clouds (MCs) offer a unique laboratory for studying galaxy interaction and the evolution of dwarf galaxies. The star formation history (SFH), which traces when and how stars formed, provides powerful constraints for the dynamical modelling of the system's past interactions and the processes of stripping and triggered star formation in tidally influenced environments. We aim to reconstruct the SFH of the Magellanic Bridge, the gaseous and stellar stream connecting the Magellanic Clouds. We used data from the deep optical STEP survey, which covers 54deg^2^ across the Small Magellanic Cloud (SMC) and the Bridge, reaching stars below the oldest main-sequence turnoff at the distance of the MCs. We applied the synthetic colour-magnitude diagram technique to 14deg^2^ of STEP data. We constructed two libraries of synthetic stellar populations based on the PARSEC-COLIBRI and BaSTI stellar evolutionary models, with metallicities in the range -2.0<=[Fe/H]<=0 across the whole Hubble time. We find a clear peak of recent star formation ~100Myr ago in the Magellanic Bridge, which becomes increasingly pronounced towards the SMC. The low metallicity of this population suggests that it formed from gas stripped from the SMC during its most recent close encounter with the LMC. In the eastern part of the Bridge (LMC side), star formation peaks at earlier times, around 10Gyr and 2Gyr ago. We estimate a total stellar mass in the Bridge of (5.1+/-0.2)x10^5^M_{sun}_ and a present-day stellar metallicity of [$Fe/H]~-0.6dex, close to SMC value.