We present a quasielastic neutron scattering (QENS) study of single-particle dynamics in pure water, measured at temperatures between 256 and 293 K along an isobaric path at 200 MPa. A thorough analysis of the spectral line shapes reveals a departure from simple models of continuous or jump diffusion, with such an effect becoming stronger at lower temperatures. We show that such a diverging trend of dynamical quantities upon cooling closely resembles the divergent (anomalous) compressibility observed in water by small-angle diffraction. Such an analogy suggests an interesting interplay between single-particle diffusion and structural arrangements in liquid water, both bearing witness of the well-known water anomalies. In particular, a fit of dynamical parameters by a Vogel-Tammann-Fulcher law provides a critical temperature of about 220K, interestingly close to the hypothesized position of the second critical point of water and to the so-called Widom line.