Context. Star formation in galaxies is inefficient, and understanding how star formation is regulated in galaxies is one of the most fundamental challenges of contemporary astrophysics. Radiative cooling, feedback from supernovae and active galactic nuclei (AGN), and large-scale dynamics and dissipation of turbulent energy act over various time and spatial scales and all regulate star formation in a complex gas cycle. Aims. This paper presents the physics implemented in a new semi-analytical model of galaxy formation and evolution called the Galaxy Assembler from dark-matter Simulation (G.A.S.). Methods. The fundamental underpinning of our new model is the development of a multiphase interstellar medium (ISM) in which energy produced by supernovae and AGN maintains an equilibrium between a diffuse, hot, and stable gas and a cooler, clumpy, and low-volume filling factor gas. The hot gas is susceptible to thermal and dynamical instabilities. We include a description of how turbulence leads to the formation of giant molecular clouds through an inertial turbulent energy cascade, assuming a constant kinetic energy transfer per unit volume. We explicitly modelled the evolution of the velocity dispersion at different scales of the cascade and accounted for thermal instabilities in the hot halo gas. Thermal instabilities effectively reduce the impact of radiative cooling and moderates accretion rates onto galaxies, and in particular, for those residing in massive haloes. Results. We show that rapid and multiple exchanges between diffuse and unstable gas phases strongly regulates star formation rates in galaxies because only a small fraction of the unstable gas is forming stars. We checked that the characteristic timescales describing the gas cycle, gas depletion timescale, and star-forming laws at different scales are in good agreement with observations. For high-mass haloes and galaxies, cooling is naturally regulated by the growth of thermal instabilities, so we do not need to implement strong AGN feedback in this model. Our results are also in good agreement with the observed stellar mass function from z ≃ 6.0 to z ≃ 0.5. Conclusion. Our model offers the flexibility to test the impact of various physical processes on the regulation of star formation on a representative population of galaxies across cosmic times. Thermal instabilities and the cascade of turbulent energy in the dense gas phase introduce a delay between gas accretion and star formation, which keeps galaxy growth inefficient in the early Universe. The main results presented in this paper, such as stellar mass functions, are available in the GALAKSIENN library.