ABSTRACTUsing the modelling code x-cigale, we reproduced the spectral energy distributions (SEDs) of 1359 SDSS quasi-stellar objects (QSOs) within the redshift range 0 < z < 4, for which we have NIR/MIR fluxes with the highest quality and spectral data characterizing their supermassive black hole (SMBHs). Consistent with a rapid formation of the host galaxies, the star formation histories (SFHs) have small e-folding, at most 750 Myr using an SFH function for spiral or 1000 Myr using one for elliptical. Above z ∼ 1.6, the two solutions are degenerate, the SEDs being dominated by the active galactic nucleus (AGN) continuum and high star formation rates (SFRs), typical of starburst galaxies, while at lower redshifts the starburst nature of the host, independent from its morphology, is better reproduced by a spiral SFH. In general, the SFR increases with the redshift, the mass of the bulge, the AGN luminosity, and Eddington ratio, suggesting there is no evidence of AGN quenching of star formation. Comparing the specific BH accretion rate (BHAR) with specific SFR, all the QSOs at any redshift trace a linear sequence below the Eddington luminosity, in parallel and above the one-to-one relation, implying that QSOs are in a special phase of evolution during which the growth in mass of their SMBH is more rapid than the growth in mass of their galaxy hosts. This particular phase is consistent with a scenario where the galaxy hosts of QSOs in the past grew in mass more rapidly than their SMBHs, suggesting that a high star formation efficiency during their formation was responsible in limiting their masses.