  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 $z\sim $0.5-2.5 from the literature. We find that the resolved
velocity dispersion is systematically lower by a factor $2.45\pm0.38$ 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.
