Nova explosions are caused by global thermonuclear runaways triggered in the surface layers of accreting white dwarfs1–3. It has been predicted4–6 that localized thermonuclear bursts on white dwarfs can also take place, similar to type-I X-ray bursts observed in accreting neutron stars. Unexplained rapid bursts from the binary system TV Columbae, in which mass is accreted onto a moderately strong magnetized white dwarf from a low-mass companion, have been observed on several occasions in the past 40 years7–11. During these bursts, the optical/ultraviolet luminosity increases by a factor of more than three in less than an hour and fades in around ten hours. Fast outflows have been observed in ultraviolet spectral lines7, with velocities of more than 3,500 kilometres per second, comparable to the escape velocity from the white dwarf surface. Here we report on optical bursts observed in TV Columbae and in two additional accreting systems, EI Ursae Majoris and ASASSN-19bh. The bursts have a total energy of approximately 10−6 times than those of classical nova explosions (micronovae) and bear a strong resemblance to type-I X-ray bursts12–14. We exclude accretion or stellar magnetic reconnection events as their origin and suggest thermonuclear runaway events in magnetically confined accretion columns as a viable explanation.