Starburst galaxies at the peak of cosmic star formation are among the most extreme star-forming engines in the Universe, producing stars over about 100 million years (ref. 2). The star-formation rates of these galaxies, which exceed 100 solar masses per year, require large reservoirs of cold molecular gas to be delivered to their cores, despite strong feedback from stars or active galactic nuclei. Consequently, starburst galaxies are ideal for studying the interplay between this feedback and the growth of a galaxy. The methylidyne cation, CH+, is a most useful molecule for such studies because it cannot form in cold gas without suprathermal energy input, so its presence indicates dissipation of mechanical energy or strong ultraviolet irradiation. Here we report the detection of CH+ (J = 1–0) emission and absorption lines in the spectra of six lensed starburst galaxies at redshifts near 2.5. This line has such a high critical density for excitation that it is emitted only in very dense gas, and is absorbed in low-density gas. We find that the CH+ emission lines, which are broader than 1,000 kilometres per second, originate in dense shock waves powered by hot galactic winds. The CH+ absorption lines reveal highly turbulent reservoirs of cool (about 100 kelvin), low-density gas, extending far (more than 10 kiloparsecs) outside the starburst galaxies (which have radii of less than 1 kiloparsec). We show that the galactic winds sustain turbulence in the 10-kiloparsec-scale environments of the galaxies, processing these environments into multiphase, gravitationally bound reservoirs. However, the mass outflow rates are found to be insufficient to balance the star-formation rates. Another mass input is therefore required for these reservoirs, which could be provided by ongoing mergers or cold-stream accretion. Our results suggest that galactic feedback, coupled jointly to turbulence and gravity, extends the starburst phase of a galaxy instead of quenching it.