Hazardous tsunamis are known to be generated predominantly at subduction zones by large earthquakes on dip (vertical)-slip faults. However, a moment magnitude ($M_{w}$) 7.5 earthquake on a strike (lateral)-slip fault in Sulawesi (Indonesia) in 2018 generated a tsunami that devastated the city of Palu. The mechanism by which this large tsunami originated from a strike-slip earthquake has been debated. Here we present near-field ground motion data from a GPS station that confirms that the 2018 Palu earthquake attained supershear speed, i.e., a rupture speed greater than the speed of shear waves in the host medium. We study the effect of this supershear rupture on tsunami generation by coupling the ground motion to a 1D non-linear shallow-water wave model that accounts for both the time-dependent bathymetric displacement and velocity. With the local bathymetric profile of the Palu bay around a tidal gauge, we find that these simulations reproduce the tsunami motions measured by the gauge, with only minimal tuning of parameters. We conclude that Mach (shock) fronts, generated by the supershear speed of the earthquake, interacted with the bathymetry and contributed to the tsunami. This suggests that rupture speed should be considered in tsunami hazard assessments.