ABSTRACT We present results from spectroscopic observations with X-shooter at the Very Large Telescope of seven H2-bearing damped Lyman-α systems (DLAs) at high redshifts (zabs ∼ 2.5–3). These DLAs were originally selected from the presence of strong H2 lines directly seen at the DLA redshift in low-resolution low signal-to-noise ratio SDSS spectra. We confirm the detection of molecular hydrogen in all of them. We measure the column densities of H i, H2 in various rotational levels, and metal species, and associated dust extinction. The metallicities, obtained from undepleted species, are in the range log Z = −0.8 to −0.2. We discuss the chemical enrichment in these clouds and compare their properties with that of other molecular-rich systems selected by other means. In particular, we show that three different methods of pre-selection of H2-bearing DLAs in the SDSS have their own biases but complement each other mostly in terms of chemical enrichment. We use the rotational excitation of H2 molecules together with the fine-structure energy levels of neutral carbon to constrain the physical conditions in the gas with the help of numerical modelling as well as analytical expressions for the surface density at which atomic to molecular conversion happens. We find that the H2-bearing medium revealed by the studied DLAs has typical values for the kinetic temperature, hydrogen density, and UV radiation field of T ∼ 100 K, nH ∼ 100 cm−3, and IUV , respectively, about twice the intensity of the Draine field. Detailed studies combining different selections should therefore bring important clues to understand the H i-H2 transition at high redshift.