The density and temperature dependence of the structural relaxation time (tau) in water was determined by inelastic ultraviolet scattering spectroscopy in the thermodynamic range (P=1-4000 bars, T=253-323 K), where several water anomalies take place. We observed an activation (Arrhenius) temperature dependence of tau at constant density and a monotonic density decrease at constant temperature. The latter trend was accounted for by introducing a density-dependent activation entropy associated to water local structure. The combined temperature and density behavior of tau indicates that differently from previous results, in the probed thermodynamic range, the relaxation process is ruled by a density-dependent activation Helmholtz free energy rather than a simple activation energy. Finally, the extrapolation of the observed phenomenology at lower temperature suggests a substantial agreement with the liquid-liquid phase transition hypothesis.