We present an analysis of the kinematics and excitation of the warm ionized gas in two obscured, powerful quasars at z≥ 3.5 from the Spitzer Wide-area Infrared Extragalactic (SWIRE) survey, SWIRE J022513.90−043419.9 and SWIRE J022550.67−042142, based on imaging spectroscopy on the Very Large Telescope. Line ratios in both targets are consistent with luminous narrow-line regions of active galactic nuclei (AGN). SWIRE J022550.67−042142 has very broad [full width at half-maximum (FWHM) = 2000 km s−1], spatially compact [O iii] line emission. SWIRE J022513.90−043419.9 is spatially resolved, has complex line profiles of Hβ and [O iii], including broad wings with blueshifts of up to −1500 km s−1 relative to the narrow [O iii] λ5007 component, and widths of up to FWHM = 5000 km s−1. Estimating the systemic redshift from the narrow Hβ line, as is standard for AGN host galaxies, implies that a significant fraction of the molecular gas is blueshifted relative to the systemic velocity. Thus the molecular gas could be participating in the outflow. Significant fractions of the ionized and molecular gas reach velocities greater than the escape velocity. We compare empirical and modelling constraints for different energy injection mechanisms, such as merging, star formation and momentum-driven AGN winds. We argue that the radio source is the most likely culprit, in spite of the source's rather modest radio power of 1025 W Hz−1. Such a radio power is not uncommon for intense starburst galaxies at z∼ 2. We discuss these results in light of the co-evolution of AGN and their host galaxy.