Can I kill bacteria with distilled water?
Indeed, bacteria are susceptible to osmotic stress from being in pure water. They are also susceptible to starvation in such an environment.

However, wild bacteria tend to have many mechanisms that protect against this and allow them to survive in spite of being exposed to water. Note, also, that freshwater can serve as a habitat for many species of bacteria: Consider ponds, streams, lakes, etc.

Thus bacteria which you encounter in everyday life will likely have been those that survived osmotic stress in the first place. And drying, and mild disinfectants, and sunlight, and temperature variation... So it is unlikely that water alone will kill them.

Here is a recent article on this topic: http://jgp.rupress.org/content/early/2015/04/07/jgp.201411296.short

And here's one about how osmotic stress affects susceptibility to other antibacterial effects: https://www.sciencedirect.com/science/article/pii/S0740002014001890