We investigate the evolution of the H β + O iii and O ii luminosity functions from z ∼ 0.8 to ∼5 in four redshift slices per emission line using data from the High-z Emission Line Survey (HiZELS). This is the first time that the H β + O iii and O ii luminosity functions have been studied at these redshifts in a self-consistent analysis. This is also the largest sample of O ii and H β + O iii emitters (3475 and 3298 emitters, respectively) in this redshift range, with large comoving volumes ∼1 × 106 Mpc−3 in two independent volumes (COSMOS and UDS), greatly reducing the effects of cosmic variance. The emitters were selected by a combination of photometric redshift and colour–colour selections, as well as spectroscopic follow-up, including recent spectroscopic observations using DEIMOS and MOSFIRE on the Keck Telescopes and FMOS on Subaru. We find a strong increase in L ⋆ and a decrease in ϕ⋆ for both H β + O iii and O ii emitters. We derive the O ii star formation history of the Universe since z ∼ 5 and find that the cosmic star formation rate density (SFRD) rises from z ∼ 5 to ∼3 and then drops towards z ∼ 0. We also find that our star formation history is able to reproduce the evolution of the stellar mass density up to z ∼ 5 based only on a single tracer of star formation. When comparing the H β + O iii SFRDs to the O ii and H α SFRD measurements in the literature, we find that there is a remarkable agreement, suggesting that the H β + O iii sample is dominated by star-forming galaxies at high-z rather than AGNs.