We present Atacama Large Millimeter/submillimeter Array (ALMA) observations of six radio-loud quasar host galaxies at z = 1.4-2.3. We combine the kiloparsec-scale resolution ALMA observations with high spatial resolution adaptive optics integral field spectrograph data of the ionized gas. We detect molecular gas emission in five quasar host galaxies and resolve the molecular interstellar medium using the CO (3-2) or CO (4-3) rotational transitions. Clumpy molecular outflows are detected in four quasar host galaxies and a merger system 21 kpc away from one quasar. Between the ionized and cold molecular gas phases, the majority of the outflowing mass is in a molecular phase, while for three out of four detected multiphase gas outflows, the majority of the kinetic luminosity and momentum flux is in the ionized phase. Combining the energetics of the multiphase outflows, we find that their driving mechanism is consistent with energy-conserving shocks produced by the impact of the quasar jets with the gas in the galaxy. By assessing the molecular gas mass to the dynamics of the outflows, we estimate a molecular gas depletion timescale of a few megayears. The gas outflow rates exceed the star formation rates, suggesting that quasar feedback is a major mechanism of gas depletion at the present time. The coupling efficiency between the kinetic luminosity of the outflows and the bolometric luminosity of the quasar of 0.1%-1% is consistent with theoretical predictions. Studying multiphase gas outflows at high redshift is important for quantifying the impact of negative feedback in shaping the evolution of massive galaxies.