Despite observed strong correlations between central supermassive black holes (SMBHs) and star formation in galactic nuclei, uncertainties exist in our understanding of their coupling. We present observations of the ratio of heavily obscured to unobscured quasars as a function of cosmic epoch up to z congruent with 3 and show that a simple physical model describing mergers of massive, gas-rich galaxies matches these observations. In the context of this model, every obscured and unobscured quasar represents two distinct phases that result from a massive galaxy merger event. Much of the mass growth of the SMBH occurs during the heavily obscured phase. These observations provide additional evidence for a causal link between gas-rich galaxy mergers, accretion onto the nuclear SMBH, and coeval star formation.