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Predicting the star formation rate (SFR) in galaxies is crucial to understand their evolution and morphology. To do so requires a fine understanding of how dense structures of gas are created and collapse. In that, turbulence and gravity play a major role. Within the gravo-turbulent framework, we assume that turbulence shapes the ISM, creating density fluctuations that, if gravitationally unstable, will collapse and form stars. The goal of this work is to quantify how different regimes of turbulence, characterized by the strength and compressibility of the driving, shape the density field. We are interested in the outcome in terms of SFR and how it compares with existing analytical models for the SFR. We run a series of hydrodynamical simulations of turbulent gas. The simulations are first conducted without gravity, so that the density and velocity are shaped by the turbulence driving. Gravity is then switched on, and the SFR is measured and compared with analytical models. The physics included in these simulations is very close to the one assumed in the classical gravo-turbulent SFR analytical models, which makes the comparison straightforward. We found that the existing analytical models convincingly agree with simulations at low Mach number, but we measure a much lower SFR in the simulation with a high Mach number. We develop, in a companion paper, an updated physically-motivated SFR model that reproduces well the inefficient high Mach regime of the simulations. Our work demonstrates that accurate estimations of the turbulent-driven replenishment time of dense structures and the dense gas spatial distribution are necessary to correctly predict the SFR in the high Mach regime. The inefficient high-Mach regime is a possible explanation for the low SFR found in dense and turbulent environments such as the centers of our Milky Way and other galaxies., Comment: 13 pages, 14 figures, accepted for publication in A&A