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We examine the distribution of radio emission from ∼42 000 quasars from the Sloan Digital Sky Survey, as measured in the LOFAR Two-metre Sky Survey (LoTSS). We present a model of the radio luminosity distribution of the quasars that assumes that every quasar displays a superposition of two sources of radio emission: active galactic nuclei (jets) and star formation. Our two-component model provides an excellent match to the observed radio flux density distributions across a wide range of redshifts and quasar optical luminosities; this suggests that the jet-launching mechanism operates in all quasars but with different powering efficiency. The wide distribution of jet powers allows for a smooth transition between the ‘radio-quiet’ and ‘radio-loud’ quasar regimes, without need for any explicit bimodality. The best-fitting model parameters indicate that the star formation rate of quasar host galaxies correlates strongly with quasar luminosity and also increases with redshift at least out to z ∼ 2. For a model where star formation rate scales as Lαbol(1+z)β, we find α = 0.47 ± 0.01 and β = 1.61 ± 0.05, in agreement with far-infrared studies. Quasars contribute ≈0.15 per cent of the cosmic star formation rate density at z = 0.5, rising to 0.4 per cent by z ∼ 2. The typical radio jet power is seen to increase with both increasing optical luminosity and black hole mass independently, but does not vary with redshift, suggesting intrinsic properties govern the production of the radio jets. We discuss the implications of these results for the triggering of quasar activity and the launching of jets.