We introduce a new set of large scale, high resolution hydrodynamical simulations of the intergalactic medium: the Sherwood simulation suite. These are performed in volumes 10^3-160^3 h^-3 comoving Mpc^{3}, span almost four orders of magnitude in mass resolution with up to 17.2 billion particles, and employ a variety of physics variations including warm dark matter and galactic outflows. We undertake a detailed comparison of the simulations to high resolution, high signal-to-noise observations of the Lyman-alpha forest over the redshift range 2 < z < 5. The simulations are in very good agreement with the observational data, lending further support to the paradigm that the Lyman-alpha forest is a natural consequence of the web-like distribution of matter arising in LCDM cosmological models. Only a small number of minor discrepancies remain with respect to the observational data. Saturated Lyman-alpha absorption lines with column densities N_HI > 10^14.5 cm^-2 at 2 < z < 2.5 are underpredicted in the models. An uncertain correction for continuum placement bias is required to match the distribution and power spectrum of the transmitted flux, particularly at z > 4. Finally, the temperature of intergalactic gas in the simulations may be slightly too low at z=2.7 and a flatter temperature-density relation is required at z=2.4, consistent with the expected effects of non-equilibrium ionisation during He-II reionisation.