We use the HiZELS narrow-band Hα survey in combination with CANDELS, UKIDSS and WIRDS near-infrared imaging, to investigate the morphologies, merger rates and sizes of a sample of Hα emitting galaxies in the redshift range z = 0.40-2.23, an epoch encompassing the rise to the peak of the star formation rate density. Merger rates are estimated from space- and ground-based imaging using the M₂₀ coefficient. To account for the increase in the specific star formation rate (sSFR) of the star forming 'main sequence' with redshift, we normalize the star formation rates of galaxies at each epoch to the typical value derived from the Hα luminosity function. Once this trend in sSFR is removed we see no evidence for an increase in the number density of star-forming galaxies or the merger rate with redshift. We thus conclude that neither is the main driver of the enhanced star-formation rate density at z ∼ 1-2, with secular processes such as instabilities within efficiently fuelled, gas-rich discs or multiple minor mergers the most likely alternatives. However, we find that ∼40-50 per cent of starburst galaxies, those with enhanced specific star formation at their epoch, are major mergers and this fraction is redshift independent. Finally, we find the surprising result that the typical size of a star-forming galaxy of a given mass does not evolve across the redshift range considered, suggesting a universal size-mass relation. Taken in combination, these results indicate a star-forming galaxy population that is statistically similar in physical size, merger rate and mass over the ∼6 Gyr covered in this study, despite the increase in typical sSFR.