We present deep echelle spectrophotometry of the brightest emission-line knots of the star-forming galaxies He 2-10, Mrk 1271, NGC 3125, NGC 5408, POX 4, SDSS J1253-0312, Tol 1457-262, Tol 1924-416 and the H II region Hubble V in the Local Group dwarf irregular galaxy NGC 6822. The data have been taken with the Very Large Telescope Ultraviolet-Visual Echelle Spectrograph in the 3100-10420 Å range. We determine electron densities and temperatures of the ionized gas from several emission-line intensity ratios for all the objects. We derive the ionic abundances of C2+ and/or O2+ from faint pure recombination lines in several of the objects, permitting to derive their C/H and C/O ratios. We have explored the chemical evolution at low metallicities analysing the C/O versus O/H, C/O versus N/O and C/N versus O/H relations for Galactic and extragalactic H II regions and comparing with results for Galactic halo stars and damped Lyα systems. We find that H II regions in star-forming dwarf galaxies occupy a different locus in the C/O versus O/H diagram than those belonging to the inner discs of spiral galaxies, indicating their different chemical evolution histories, and that the bulk of C in the most metal-poor extragalactic H II regions should have the same origin than in halo stars. The comparison between the C/O ratios in H II regions and in stars of the Galactic thick and thin discs seems to give arguments to support the merging scenario for the origin of the Galactic thick disc. Finally, we find an apparent coupling between C and N enrichment at the usual metallicities determined for H II regions and that this coupling breaks in very low metallicity objects.