  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.
