Context. Ionized outflows are ubiquitous in non-radio-loud obscured quasars (QSO2s) at different redshifts. However, the actual size of the outflows and their efficiency for gas ejection and star formation truncation are controversial. Large-scale (exceeding several kiloparsec) extended radio structures might be necessary to identify (even to trigger) outflow signatures across such large spatial scales. Aims: We search for large-scale ionized outflows associated with six optically selected QSO2 (five non-radio-loud and one radio-loud) at z ˜ 0.2-0.5, targeting objects with extended radio structures. We also investigate the dynamical state of the QSO2 host galaxies. Methods: We obtained data with the optical imager and long-slit spectrograph (OSIRIS) mounted on the 10.4m Gran Telescopio Canarias Spanish telescope (GTC) for these six QSO2 with the slit located along the radio axis. We traced the gas kinematics with the [OIII]λλ4959,5007 lines to investigate ionized outflows and characterize the dynamical state of the host galaxies. This second study was complemented with previously published spectroscopic data obtained with the multimode focal reducer and low dispersion spectrograph (FORS2) mounted on the Very Large Telescope (VLT) of 13 more QSO2 at similar z. Results: We identify ionized outflows in four out of the six QSO2 observed with the GTC. The outflows are spatially unresolved in two QSO2 and compact in a third (radial size of R = 0.8 ± 0.3 kpc). Of particular interest is the radio-quiet QSO2 SDSS 0741+3020 at z = 0.47. It is associated with a giant ˜112 kpc nebula. An ionized outflow probably induced by the radio structures has been detected along the axis defined by the central ˜1″ radio structure, extending up to at least ˜4 kpc from the active galactic nucleus (AGN). Turbulent gas (σ ˜ 130 km s-1) has also been detected across the giant gas nebula up to ˜40 kpc from the AGN. This turbulence may have been induced by outflows triggered by the interaction between a so-far undetected large-scale radio source and the nebula. Regarding the dynamical state of the host galaxies, we find that the majority of the QSO2 show v/σ < 1, implying that they are dominated by random motions (so-called dispersion-dominated systems). Most (17 of 19) fall in the area of the E/S0 galaxies in the dynamical diagram v/σ versus σ. None are consistent with spiral or disk galaxies.