Active galactic nuclei (AGN) feedback operated by the expansion of radio jets can play a crucial role in driving gaseous outflows on galaxy scales. Galaxies hosting young radio AGN, whose jets are in the first phases of expansion through the surrounding interstellar medium (ISM), are the ideal targets to probe the energetic significance of this mechanism. In this paper, we characterise the warm ionised gas outflows in a sample of nine young radio sources from the 2 Jy sample, combining X-shooter spectroscopy and Hubble Space Telescope imaging data. We find that the warm outflows have similar radial extents (similar to 0.06-2 kpc) as radio sources, consistent with the idea that "jet mode" AGN feedback is the dominant driver of the outflows detected in young radio galaxies. Exploiting the broad spectral coverage of the X-shooter data, we used the ratios of trans-auroral emission lines of [SII] and [OII] to estimate the electron densities, finding that most of the outflows have gas densities (log(n(e) cm(-3)) similar to 3-4.8), which we speculate could be the result of compression by jet-induced shocks. Combining our estimates of the emission-line luminosities, radii, and densities, we find that the kinetic powers of the warm outflows are a relatively small fraction of the energies available from the accretion of material onto the central supermassive black hole, reflecting AGN feedback efficiencies below 1% in most cases. Overall, the warm outflows detected in our sample are strikingly similar to those found in nearby ultraluminous infrared galaxies, but more energetic and with higher feedback efficiencies on average than the general population of nearby AGN of similar bolometric luminosity; this is likely to reflect a high degree of coupling between the jets and the near-nuclear ISM in the early stages of radio source evolution.