We study the properties of interstellar medium (ISM) substructure and turbulence in hydrodynamic [adaptive mesh refinement (AMR)] galaxy simulations with resolutions up to 0.8 pc and 5 × 103 Mȯ. We analyse the power spectrum of the density distribution, and various components of the velocity field. We show that the disc thickness is about the average Jeans scalelength, and is mainly regulated by gravitational instabilities. From this scale of energy injection, a turbulence cascade towards small scale is observed, with almost isotropic small-scale motions. On scales larger than the disc thickness, density waves are observed, but there is also a full range of substructures with chaotic and strongly non-isotropic gas velocity dispersions. The power spectrum of vorticity in a Large Magellanic Cloud sized model suggests that an inverse cascade of turbulence might be present, although energy input over a wide range of scales in the coupled gaseous+stellar fluid could also explain this quasi-two-dimensional regime on scales larger than the disc scaleheight. Similar regimes of gas turbulence are also found in massive high-redshift discs with high gas fractions. Disc properties and ISM turbulence appear to be mainly regulated by gravitational processes, both on large scales and inside dense clouds. Star formation feedback is however essential to maintain the ISM in a steady state by balancing a systematic gas dissipation into dense and small clumps. Our galaxy simulations employ a thermal model based on a barotropic equation of state aimed at modelling the equilibrium of gas between various heating and cooling processes. Denser gas is typically colder in this approach, which is shown to correctly reproduce the density structures of a star-forming, turbulent, unstable and cloudy ISM down to scales of a few parsecs.