  Massive starburst galaxies in the early Universe are estimated to have depletion times of $\sim 100$ Myr and thus be able to convert their gas very quickly into stars, possibly leading to a rapid quenching of their star formation. For these reasons, they are considered progenitors of massive early-type galaxies (ETGs). In this paper, we study two high-$z$ starbursts, AzTEC/C159 ($z\simeq 4.57$) and J1000+0234 ($z\simeq 4.54$), observed with ALMA in the [CII] 158-$\mu$m emission line. These observations reveal two massive and regularly rotating gaseous discs. A 3D modelling of these discs returns rotation velocities of about $500$ km/s and gas velocity dispersions as low as $\approx 20$ km/s, leading to very high ratios between regular and random motion ($V/\sigma {\lower.7ex\hbox{$\;\stackrel{\textstyle>}{\sim}\;$}} 20$), at least in AzTEC/C159. The mass decompositions of the rotation curves show that both galaxies are highly baryon-dominated with gas masses of $\approx 10^{11}M_{\odot}$, which, for J1000+0234, is significantly higher than previous estimates. We show that these high-$z$ galaxies overlap with $z=0$ massive ETGs in the ETG analogue of the stellar-mass Tully-Fisher relation once their gas is converted into stars. This provides dynamical evidence of the connection between massive high-$z$ starbursts and ETGs, although the transformation mechanism from fast rotating to nearly pressure-supported systems remains unclear. 
Fast rotating and low-turbulence discs at $z\simeq 4.5$: Dynamical   evidence of their evolution into local early-type galaxies