The Schmidt-Kennicutt relation links the surface densities of gas to the star formation rate in galaxies. The physical origin of this relation, and in particular its break, i.e., the transition between an inefficient regime at low gas surface densities and a main regime at higher densities, remains debated. Here, we study the physical origin of the star formation relations and breaks in several low-redshift galaxies, from dwarf irregulars to massive spirals. We use numerical simulations representative of the Milky Way and the Large and Small Magellanic Clouds with parsec up to subparsec resolution, and which reproduce the observed star formation relations and the relative variations of the star formation thresholds. We analyze the role of interstellar turbulence, gas cooling, and geometry in drawing these relations at 100 pc scale. We suggest in particular that the existence of a break in the Schmidt-Kennicutt relation could be linked to the transition from subsonic to supersonic turbulence and is independent of self-shielding effects. With this transition being connected to the gas thermal properties and thus to the metallicity, the break is shifted toward high surface densities in metal-poor galaxies, as observed in dwarf galaxies. Our results suggest that together with the collapse of clouds under self-gravity, turbulence (injected at galactic scale) can induce the compression of gas and regulate star formation. © 2014. The American Astronomical Society. All rights reserved..