  (Abridged). In this paper, we present results from a large set of numerical
simulations that demonstrate that H2 formation occurs rapidly in turbulent gas.
Starting with purely atomic hydrogen, large quantities of molecular hydrogen
can be produced on timescales of 1 -- 2 Myr, given turbulent velocity
dispersions and magnetic field strengths consistent with observations.
Moreover, as our simulations underestimate the effectiveness of H2
self-shielding and dust absorption, we can be confident that the molecular
fractions that we compute are strong lower limits on the true values. The
formation of large quantities of H2 on the timescale required by rapid cloud
formation models therefore appears to be entirely plausible.
  We also investigate the density and temperature distributions of gas in our
model clouds. We show that the density probability distribution function is
approximately log-normal, with a dispersion that agrees well with the
prediction of Padoan, Nordlund & Jones (1997). The temperature distribution is
similar to that of a polytrope, with an effective polytropic index gamma_eff
\simeq 0.8, although at low gas densities, the scatter of the actual gas
temperature around this mean value is considerable, and the polytropic
approximation does not capture the full range of behaviour of the gas.
