We explore the nature of heavily obscured quasar host galaxies at z∼2 using deep Hubble Space Telescope Wide Field Camera 3/infrared imaging of 28 dust-obscured galaxies (DOGs) to investigate the role of major mergers in driving black hole growth. The high levels of obscuration of the quasars selected for this study act as a natural coronagraph, blocking the quasar light and allowing a clear view of the underlying host galaxy. The sample of heavily obscured quasars represents a significant fraction of the cosmic mass accretion on supermassive black holes as the quasars have inferred bolometric luminosities around the break of the quasar luminosity function. We find that only a small fraction (4 per cent, at most 11–25 per cent) of the quasar host galaxies are major mergers. Fits to their surface brightness profiles indicate that 90 per cent of the host galaxies are either disc dominated, or have a significant disc. This disc-like host morphology, and the corresponding weakness of bulges, is evidence against major mergers and suggests that secular processes are the predominant driver of massive black hole growth. Finally, we suggest that the coincidence of mergers and active galactic nucleus activity is luminosity dependent, with only the most luminous quasars being triggered mostly by major mergers.