We compare the molecular and ionized gas velocity dispersion of 9 nearby turbulent disks, analogues to high-redshift galaxies, from the DYNAMO sample using new ALMA and GMOS/Gemini observations. We combine our sample with 12 galaxies at $zsim $0.5-2.5 from the literature. We find that the resolved velocity dispersion is systematically lower by a factor $2.45pm0.38$ for the molecular gas compared to the ionized gas, after correcting for thermal broadening. This offset is constant within the galaxy disks and indicates the co-existence of a thin molecular and thick ionized gas disks. This result has a direct impact on the Toomre $Q$ and pressure derived in galaxies. We obtain pressures $sim0.22$ dex lower on average when using the molecular gas velocity dispersion, $sigma_{0,mol}$. We find that $sigma_{0,mol}$ increases with gas fraction and star formation rate. We also obtain an increase with redshift and show that the EAGLE and FIRE simulations overall overestimate $sigma_{0,mol}$ at high redshift. Our results suggest that efforts to compare the kinematics of gas using ionized gas as a proxy for the total gas may overestimate the velocity dispersion by a significant amount in galaxies at the peak of cosmic star formation. When using the molecular gas as a tracer, our sample is not consistent with predictions from constant efficiency star formation models, even when including transport as a source of turbulence. Feedback models with variable star formation efficiency, $epsilon_{ff}$, and/or feedback efficiency, $p_*/m_*$, better predict our observations.
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