The speed at which an earthquake rupture propagates affects its energy balance and ground shaking impact. Dynamic models of supershear earthquakes, which are faster than the speed of shear waves, often start at subshear speed and later run faster than Eshelby's speed. Here we present robust evidence of an early and persistent supershear rupture at the sub-Eshelby speed of the 2018 magnitude 7.5 Palu, Indonesia, earthquake. Slowness-enhanced back-projection of teleseismic data provides a sharp image of the rupture process, along a path consistent with the surface rupture trace inferred by subpixel correlation of synthetic-aperture radar and satellite optical images. The rupture propagated at a sustained velocity of 4.1 km s-1 from its initiation to its end, despite large fault bends. The persistent supershear speed is further validated by seismological evidence of far-field Rayleigh Mach waves. The unusual features of this earthquake probe the connections between the rupture dynamics and fault structure. An early supershear transition could be promoted by fault roughness near the hypocentre. Steady rupture propagation at a speed unexpected in homogeneous media could result from the presence of a low-velocity damaged fault zone.