A new analysis of high-resolution data from the Atacama Large Millimeter/submillimeter Array (ALMA) for 5 luminous or ultra-luminous infrared galaxies gives a slope for the Kennicutt-Schmidt (KS) relation equal to 1.74 +0.09 −0.07 for gas surface densities Σ mol > 10 3 M pc −2 and an assumed constant CO-to-H 2 conversion factor. The velocity dispersion of the CO line, σ v , scales approximately as the inverse square root of Σ mol , making the empirical gas scale height determined from H ∼ 0.5σ 2 /(πGΣ mol ) nearly constant, 150-190 pc, over 1.5 orders of magnitude in Σ mol . This constancy of H implies that the average midplane density, which is presumably dominated by CO-emitting gas for these extreme star-forming galaxies, scales linearly with the gas surface density, which, in turn, implies that the gas dynamical rate (the inverse of the free-fall time) varies with Σ 1/2 mol , thereby explaining most of the super-linear slope in the KS relation. Consistent with these relations, we also find that the mean efficiency of star formation per free-fall time is roughly constant, 5%-7%, and the gas depletion time decreases at high Σ mol , reaching only ∼ 16 Myr at Σ mol ∼ 10 4 M pc −2 . The variation of σ v with Σ mol and the constancy of H are in tension with some feedback-driven models, which predict σ v to be more constant and H to be more variable. However, these results are consistent with simulations in which large-scale gravity drives turbulence through a feedback process that maintains an approximately constant Toomre Q instability parameter.