To understand the turbulence-chemistry interaction of lean hydrogen-air premixed flames at elevated pressures, we conduct a series of high-fidelity direct numerical simulations with detailed chemistry and transport by increasing pressure from 1 up to 7 atm. For a fixed ratio of the root-mean-square turbulent velocity fluctuation to the laminar flame speed, two sets of simulations were conducted: one for a fixed ratio of integral length scale to the laminar flame thickness and the other one for a fixed integral length scale. We observe that the turbulent flame speed and volume-integrated heat release rate (HRR) increase with pressure due to the promoted chemical effects. To elucidate the enhanced chemical effects, we conduct a budget analysis of diffusive transport and reaction rate of hydrogen and examine the conditionally-averaged flame structure. As opposed to the laminar flames, the diffusive effects of hydrogen for turbulent flames are found to increase with pressure in a more non-linear way, which significantly enhances the radical pool generation, leading to abundant H radical that is required for the rate of the pressure-sensitive reaction H + O2(+M) = HO2(+M) to be accelerated at higher pressures. Due to the strong diffusive transport of hydrogen for pressurized turbulence, not only does another small HRR peak emerge upstream, but also the location of the main HRR peak is moved towards the upstream region. Furthermore, the probability density function (PDF) of the flame curvature, computed for the isosurface associated with maximal diffusive transport of hydrogen, shows a clear trend of the PDF peak shifting to larger positive values as pressure increases. Simulations with the inclusion of thermal diffusion are also carried out at atmospheric and elevated pressure conditions to quantify the Soret effect. It is found that the magnitude of the hydrogen diffusion rate is augmented by thermal diffusion, affecting its mass fraction distribution. The mean turbulent flame speed is enhanced by approximately 8% and 7% for the 1 and 7 atm conditions, respectively.