We study the relaxation times tα in the water-methanol system. We examine new data and data from the literature in the large temperature range 163 < T < 335 K obtained using different experimental techniques and focus on how tα affects the hydrogen bond structure of the system and the hydrophobicity of the alcohol methyl group. We examine the relaxation times at a fixed temperature as a function of the water molar fraction XW and observe two opposite behaviors in their curvature when the system moves from high to low T regimes. This behavior differs from that of an ideal solution in that it has excess values located at different molar fractions (XW = 0.5 for high T and 0.75 in the deep supercooled regime). We analyze the data and find that above a crossover temperature T ∼ 223 K, hydrophobicity plays a significant role and below it the water tetrahedral network dominates. This temperature is coincident with the fragile-to-strong dynamical crossover observed in confined water and supports the liquid-liquid phase transition hypothesis. At the same time, the reported data suggest that this crossover temperature (identified as the Widom line temperature) also depends on the alcohol concentration.