The deep connection between galaxies and their supermassive black holes is central to modern astrophysics and cosmology. The observed correlation between galaxy and black hole mass is usually attributed to the contribution of major mergers to both. We make use of a sample of galaxies whose disc-dominated morphologies indicate a major-merger-free history and show that such systems are capable of growing supermassive black holes at rates similar to quasars. Comparing black hole masses to conservative upper limits on bulge masses, we show that the black holes in the sample are typically larger than expected if processes creating bulges are also the primary driver of black hole growth. The same relation between black hole and total stellar mass of the galaxy is found for the merger-free sample as well as a sample that has experienced substantial mergers, indicating that major mergers do not play a significant role in controlling the co-evolution of galaxies and black holes. We suggest that more fundamental processes that contribute to galaxy assembly are also responsible for black hole growth.