It is widely known that the gas in galaxy discs is highly turbulent, but there is much debate on which mechanism can energetically maintain this turbulence. Among the possible candidates, supernova (SN) explosions are likely the primary drivers but doubts remain on whether they can be sufficient in regions of moderate star formation activity, in particular in the outer parts of discs. Thus, a number of alternative mechanisms have been proposed. In this paper, we measure the SN efficiency η , namely the fraction of the total SN energy needed to sustain turbulence in galaxies, and verify that SNe can indeed be the sole driving mechanism. The key novelty of our approach is that we take into account the increased turbulence dissipation timescale associated with the flaring in outer regions of gaseous discs. We analyse the distribution and kinematics of HI and CO in ten nearby star-forming galaxies to obtain the radial profiles of the kinetic energy per unit area for both the atomic gas and the molecular gas. We use a theoretical model to reproduce the observed energy with the sum of turbulent energy from SNe, as inferred from the observed star formation rate (SFR) surface density, and the gas thermal energy. For the atomic gas, we explore two extreme cases in which the atomic gas is made either of cold neutral medium or warm neutral medium, and the more realistic scenario with a mixture of the two phases. We find that the observed kinetic energy is remarkably well reproduced by our model across the whole extent of the galactic discs, assuming η constant with the galactocentric radius. Taking into account the uncertainties on the SFR surface density and on the atomic gas phase, we obtain that the median SN efficiencies for our sample of galaxies are ⟨ η atom ⟩ = 0.015 −0.008 +0.018 for the atomic gas and ⟨ η mol ⟩ = 0.003 −0.002 +0.006 for the molecular gas. We conclude that SNe alone can sustain gas turbulence in nearby galaxies with only few percent of their energy and that there is essentially no need for any further source of energy.