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 ). The initial dataset comprises 17 galaxy disks with high data quality from the ALPAKA sample, supplemented with galaxies from the literature, resulting in a sample of 57 galaxy disks spanning the redshift range $z = 0 - 5$. This extended sample consists of main-sequence and starburst galaxies with stellar masses $ M_ odot $. The comparison with current Halpha kinematic observations and existing models demonstrates that the velocity dispersion inferred from cold gas tracers differ by a factor of $ 3$ from those obtained using emission lines tracing the warm, ionized gas. We show that stellar feedback is the main driver of turbulence measured from cold gas tracers and the physics of turbulence driving does not appear to evolve with time. 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.