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We use the first systematic data sets of CO molecular line emission in z∼ 1–3 normal star-forming galaxies (SFGs) for a comparison of the dependence of galaxy-averaged star formation rates on molecular gas masses at low and high redshifts, and in different galactic environments. Although the current high- z samples are still small and biased towards the luminous and massive tail of the actively star-forming ‘main-sequence’, a fairly clear picture is emerging. Independent of whether galaxy-integrated quantities or surface densities are considered, low- and high- z SFG populations appear to follow similar molecular gas–star formation relations with slopes 1.1 to 1.2, over three orders of magnitude in gas mass or surface density. The gas-depletion time-scale in these SFGs grows from 0.5 Gyr at to 1.5 Gyr at . The average corresponds to a fairly low star formation efficiency of 2 per cent per dynamical time. Because star formation depletion times are significantly smaller than the Hubble time at all redshifts sampled, star formation rates and gas fractions are set by the balance between gas accretion from the halo and stellar feedback. In contrast, very luminous and ultraluminous, gas-rich major mergers at both low and high z produce on average four to 10 times more far-infrared luminosity per unit gas mass. We show that only some fraction of this difference can be explained by uncertainties in gas mass or luminosity estimators; much of it must be intrinsic. A possible explanation is a top-heavy stellar mass function in the merging systems but the most likely interpretation is that the star formation relation is driven by global dynamical effects. For a given mass, the more compact merger systems produce stars more rapidly because their gas clouds are more compressed with shorter dynamical times, so that they churn more quickly through the available gas reservoir than the typical normal disc galaxies. When the dependence on galactic dynamical time-scale is explicitly included, disc galaxies and mergers appear to follow similar gas-to-star formation relations. The mergers may be forming stars at slightly higher efficiencies than the discs. [ABSTRACT FROM AUTHOR]