We investigate the evolution of the H$β$+[OIII] and [OII] luminosity functions from $z ∼ 0.8$ to $∼5$ in four redshift slices per emission line using data from the High-{\it z} Emission Line Survey (HiZELS). This is the first time that the H$β$+[OIII] and [OII] luminosity functions have been studied at these redshifts in a self-consistent analysis. This is also the largest sample of [OII] and H$β$+[OIII] emitters (3475 and 3298 emitters, respectively) in this redshift range, with large co-moving volumes $∼ 1 \times 10^6$ 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 color-color 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$β$+[OIII] and [OII] emitters. We derive the [OII] star-formation history of the Universe since $z∼5$ and find that the cosmic 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$β$+[OIII] SFRDs to the [OII] and H$α$ SFRD measurements in the literature, we find that there is a remarkable agreement, suggesting that the H$β$+[OIII] sample is dominated by star-forming galaxies at high-$z$ rather than AGNs. ; Comment: 24 pages, 14 figures, 7 Tables, MNRAS accepted