Proteins perform specific biological functions that strongly depend on their three-dimensional structure. This three-dimensional structure, i.e. the way the protein folds, is strongly determined by the interaction between the protein and the water solvent. We study the dynamics of water in aqueous solutions of several globular proteins at different degrees of urea-induced unfolding, using polarization-resolved femtosecond infrared spectroscopy. We observe that a fraction of the water molecules is strongly slowed down by their interaction with the protein surface. By monitoring the slow water fraction we can directly probe the amount of water-exposed protein surface. We find that at mild denaturing conditions, the water-exposed surface increases by almost 50%, while the secondary structure is still intact. This finding indicates that protein unfolding starts with the protein structure becoming less tight, thereby allowing water to enter.