We present extensive simulation results on the TIP4P/2005 water model and show that it displays significantly enhanced small-angle scattering in the supercooled regime. The simulations exhibit a Widom line (TW), emanating from a liquid-liquid critical point (LLCP) in the supercooled region; TW is characterized by a maximal Ornstein-Zernike correlation length and strong small-angle scattering. The good agreement between the simulated small-angle scattering and recent experimental SAXS data [Huang et al., J. Chem. Phys. 133, 134504 (2010)] thus provides indirect evidence for the existence of a Widom line in supercooled water; both the LLCP and singularity-free (SF) scenarios are however consistent with the presence of TW. Simulations performed at 1, 1,000 and 1,500 bar show an increasing abruptness of a high-density (HDL) to low-density (LDL) liquid crossover associated with crossing TW, while simulations at 2,000 bar show a very gradual transition at lower temperatures indicating that the critical pressure (whether at T=0, as in the SF scenario, or above as in the LLCP scenario) is below 2,000 bar in this simulation model. Maxima in the isothermal compressibility and negative thermal expansion coefficient nearly coincide with TW at 1, 1,000 and 1,500 bar. Analysis of the tetrahedrality parameter Q reveals that the HDL-LDL structural transition is very sharp at 1,000 and 1,500 bar, and that structural fluctuations become strongly coupled to density fluctuations upon approaching TW. Furthermore, the tetrahedrality distribution becomes bimodal at ambient temperatures, an observation that possibly provides a link between the HDL-LDL transition and the structural bimodality in liquid water indicated by x-ray spectroscopic techniques.