We present 1-7GHz high-resolution radio imaging (VLA and e-MERLIN) and spatially resolved ionized gas kinematics for 10 z<0.2 type 2 'obscured' quasars (log[L_AGN_/erg.s^-1^]>~45) with moderate radio luminosities (log[L1.4GHz/W.Hz^-1^]=23.3-24.4). These targets were selected to have known ionized outflows based on broad [OIII] emission-line components (full width at half-maximum~800-1800km/s). Although 'radio-quiet' and not 'radio AGN' by many traditional criteria, we show that for nine of the targets, star formation likely accounts for ~<10 per cent of the radio emission. We find that ~80-90 per cent of these nine targets exhibit extended radio structures on 1-25kpc scales. The quasars' radio morphologies, spectral indices, and position on the radio size-luminosity relationship reveals that these sources are consistent with being low power compact radio galaxies. Therefore, we favour radio jets as dominating the radio emission in the majority of these quasars. The radio jets we observe are associated with morphologically and kinematically distinct features in the ionized gas, such as increased turbulence and outflowing bubbles, revealing jet-gas interaction on galactic scales. Importantly, such conclusions could not have been drawn from current low-resolution radio surveys such as FIRST. Our observations support a scenario where compact radio jets, with modest radio luminosities, are a crucial feedback mechanism for massive galaxies during a quasar phase.