ABSTRACT The discovery of galaxies with regularly rotating discs at redshifts ≥4 has been a puzzling challenge to galaxy formation models that tend to predict chaotic gas kinematics in the early Universe as a consequence of gas accretion, mergers, and efficient feedback. In this work, we investigated the kinematics of five highly resolved galaxies at z ∼ 4.5 observed with ALMA in the [C ii] 158 $μ$m emission line. The sample is diverse: AzTEC1 (starburst galaxy), BRI1335-0417 (starburst and quasar host galaxy), J081740 (normal star-forming galaxy), and SGP38326 (two starburst galaxies in a group). The five galaxies show velocity gradients, but four were found to be rotating discs, while the remaining, AzTEC1, is likely a merger. We studied the gas kinematics of the discs using 3DBAROLO and found that they rotate with maximum rotation velocities between 198 and 562 km s−1, while the gas velocity dispersions, averaged across the discs, are between 49 and 75 km s−1. The rotation curves are generally flat and the galaxies have ratios of ordered-to-random motion (V/σ) between 2.7 and 9.8. We present CANNUBI, an algorithm for fitting the disc geometry of rotating discs in 3D emission-line observations prior to modelling the kinematics, with which we find indications that these discs may have thicknesses of the order of 1 kpc. This study shows that early disc formation with a clear dominance of rotation with respect to turbulent motions is present across a variety of galaxy types.