The variability of quasars across multiple wavelengths is a useful probe of physical conditions in active galactic nuclei. In particular, variable accretion rates, instabilities, and reverberation effects in the accretion disc of a supermassive black hole are expected to produce correlated flux variations in ultraviolet (UV) and optical bands. Recent work has further argued that binary quasars should exhibit strongly correlated UV and optical periodicities. Strong UV–optical correlations have indeed been established in small samples of (N ≲ 30) quasars with well-sampled light curves, and have extended the ‘bluer-when-brighter’ trend previously found within the optical bands. Here, we further test the nature of quasar variability by examining the observed-frame UV–optical correlations among bright quasars extracted from the Half Million Quasars (HMQ) catalogue. We identified a large sample of 1315 quasars in HMQ with overlapping UV and optical light curves from the Galaxy Evolution Explorer and the Catalina Real-time Transient Survey, respectively. We find that strong correlations exist in this much larger sample, but we rule out, at ∼95 per cent confidence, the simple hypothesis that the intrinsic UV and optical variations of all quasars are fully correlated. Our results therefore imply the existence of physical mechanism(s) that can generate uncorrelated optical and UV flux variations.