We present a comparative study of molecular and ionized gas kinematics in nearby galaxies. These results are based on observations from the EDGE survey, which measured spatially resolved $^{12}$CO(J=1-0) in 126 nearby galaxies. Every galaxy in EDGE has corresponding resolved ionized gas measurements from CALIFA. Using a sub-sample of 17 rotation dominated, star-forming galaxies where precise molecular gas rotation curves could be extracted, we derive CO and H$alpha$ rotation curves using the same geometric parameters out to $gtrsim$1 $R_e$. We find that $sim$75% of our sample galaxies have smaller ionized gas rotation velocities than the molecular gas in the outer part of the rotation curve. In no case is the molecular gas rotation velocity measurably lower than that of the ionized gas. We suggest that the lower ionized gas rotation velocity can be attributed to a significant contribution from extraplanar diffuse ionized gas in a thick, turbulence supported disk. Using observations of the H$gamma$ transition also available from CALIFA, we measure ionized gas velocity dispersions and find that these galaxies have sufficiently large velocity dispersions to support a thick ionized gas disk. Kinematic simulations show that a thick disk with a vertical rotation velocity gradient can reproduce the observed differences between the CO and H$alpha$ rotation velocities. Observed line ratios tracing diffuse ionized gas are elevated compared to typical values in the midplane of the Milky Way. In galaxies affected by this phenomenon, dynamical masses measured using ionized gas rotation curves will be systematically underestimated.
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