  The gas in the interstellar medium (ISM) of galaxies is supersonically
turbulent. Measurements of turbulence typically rely on cold gas emission lines
for low-z galaxies and warm ionized gas observations for z>0 galaxies. Studies
of warm gas kinematics at z>0 conclude that the turbulence strongly evolves as
a function of redshift, due to the increasing impact of gas accretion and
mergers in the early Universe. However, recent findings suggest potential
biases in turbulence measurements derived from ionized gas at high-z, impacting
our understanding of turbulence origin, ISM physics and disk formation. We
investigate the evolution of turbulence using velocity dispersion ($\sigma$)
measurements from cold gas tracers (i.e., CO, [CI], [CII]) derived from a
sample of 57 galaxy disks spanning the redshift range z=0-5. This sample
consists of main-sequence and starburst galaxies with stellar masses $\gtrsim
10^{10} M_{\odot}$. The comparison with current H$\alpha$ kinematic
observations and existing models demonstrates that the velocity dispersion
inferred from cold gas tracers differ by a factor of $\approx 3$ from those
obtained using emission lines tracing warm gas. We show that stellar feedback
is the main driver of turbulence measured from cold gas tracers. This is
fundamentally different from the conclusions of studies based on warm gas,
which had to consider additional turbulence drivers to explain the high values
of $\sigma$. We present a model predicting the redshift evolution of turbulence
in galaxy disks, attributing the increase of $\sigma$ with redshift to the
higher energy injected by supernovae due to the elevated star-formation rate in
high-z galaxies. This supernova-driven model suggests that turbulence is lower
in galaxies with lower stellar mass compared to those with higher stellar mass.
Additionally, it forecasts the evolution of $\sigma$ in Milky-Way like
progenitors.
