The Orion Bar is the archetypal edge-on molecular cloud surface illuminated by strong ultraviolet radiation from nearby massive stars. Our relative closeness to the Orion nebula (about 1,350 light years away from Earth) means that we can study the effects of stellar feedback on the parental cloud in detail. Visible-light observations of the Orion Bar show that the transition between the hot ionized gas and the warm neutral atomic gas (the ionization front) is spatially well separated from the transition between atomic and molecular gas (the dissociation front), by about 15 arcseconds or 6,200 astronomical units (one astronomical unit is the Earth-Sun distance). Static equilibrium models used to interpret previous far-infrared and radio observations of the neutral gas in the Orion Bar (typically at 10-20 arcsecond resolution) predict an inhomogeneous cloud structure comprised of dense clumps embedded in a lower-density extended gas component. Here we report one-arcsecond-resolution millimetre-wave images that allow us to resolve the molecular cloud surface. In contrast to stationary model predictions, there is no appreciable offset between the peak of the H2 vibrational emission (delineating the H/H2 transition) and the edge of the observed CO and HCO[+] emission. This implies that the H/H2 and C[+]/C/CO transition zones are very close. We find a fragmented ridge of high-density substructures, photoablative gas flows and instabilities at the molecular cloud surface. The results suggest that the cloud edge has been compressed by a high-pressure wave that is moving into the molecular cloud, demonstrating that dynamical and non-equilibrium effects are important for the cloud evolution.