We study the H2 molecular content in high redshift damped Lyman-alpha systems (DLAs) as a function of the HI column density. We find a significant increase of the H2 molecular content around log N(HI) (cm^-2)~21.5-22, a regime unprobed until now in intervening DLAs, beyond which the majority of systems have log N(H2) > 17. This is in contrast with lines of sight towards nearby stars, where such H2 column densities are always detected as soon as log N(HI)>20.7. This can qualitatively be explained by the lower average metallicity and possibly higher surrounding UV radiation in DLAs. However, unlike in the Milky Way, the overall molecular fractions remain modest, showing that even at a large N(HI) only a small fraction of overall HI is actually associated with the self-shielded H2 gas. Damped Lyman-alpha systems with very high-N(HI) probably arise along quasar lines of sight passing closer to the centre of the host galaxy where the gas pressure is higher. We show that the colour changes induced on the background quasar by continuum (dust) and line absorption (HI Lyman and H2 Lyman & Werner bands) in DLAs with log N(HI)~22 and metallicity ~1/10 solar is significant, but not responsible for the long-discussed lack of such systems in optically selected samples. Instead, these systems are likely to be found towards intrinsically fainter quasars that dominate the quasar luminosity function. Colour biasing should in turn be severe at higher metallicities.
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