Hazardous tsunamis are known to be generated predominantly at subduction zones. However, the 2018 Mw 7.5 Palu (Indonesia) earthquake on a strike-slip fault generated a tsunami that devastated the city of Palu. The mechanism by which this tsunami originated from such an earthquake is being debated. Here we present near-field ground motion (GPS) data confirming that the earthquake attained supershear speed, i.e. a rupture speed greater than the shear wave speed of the host medium. We subsequently study the effect of this supershear rupture on tsunami generation by coupling the ground motion to a 1-D non-linear shallow-water wave model accounting for both time-dependent bathymetric displacement and velocity. With the local bathymetric profile of Palu bay around a tidal station, our simulations reproduce the tsunami arrival and motions observed by CCTV cameras. We conclude that Mach (shock) fronts, generated by the supershear speed, interacted with the bathymetry and contributed to the tsunami.