We investigate the structure and heat transport of liquid water at high pressures and temperatures, 1–50kbar and 300–600K, i.e., in a region of the phase diagram that is challenging for experimental investigations. Using equilibrium and non equilibrium molecular dynamics simulations and the TIP4P/2005 water model, we compute the structure and thermal conductivity of liquid water. At extreme pressures, 20–50kbar the tetrahedral order characteristic of the hydrogen bonded network is severely disrupted, and the liquid radial distribution function becomes very similar to that of simple liquids. At these extreme conditions the thermal conductivity does not feature an anomalous behavior, and decreases with temperature as observed in a wide range of simple liquids. The dependence of the thermal conductivity with temperature and pressure follows experimental observations, and we find that it can be accurately predicted in terms of the liquid isothermal compressibility, by using a modified Leibfried–Schlömann equation. We also analyze whether the thermal conductivity follows the T scaling behavior characteristic of hard sphere fluids, a behavior that has been suggested following the analysis of high pressure and high temperature experimental data. Upon close inspection we find clear deviations from this scaling behavior both in simulation and experimental data.