Using extensive molecular dynamics simulations combined with a novel approach to analyze the molecular displacements we analyzed the change in the dynamics above and below the crossover temperature T(x) for supercooled water. Our findings suggest that the crossover from fragile to strong glass former occurring at T(x) is related with a change in the diffusion mechanism evidencing the presence of jump-like diffusion at lower temperatures. Also we observe that fluctuations of the local environments are intimately connected with fluctuations in the size and the amount of cooperative cluster of mobile molecules, and in particular we find a highly cooperative nature of the motion at low temperatures.