There could be a number of reasons for this. Let us look at two.
First, rural areas have fewer people and lower population densities. This makes it more difficult for gang problems to arise. It also means that there are simply fewer poor people pushed together in small areas. This makes it harder for a general culture of crime to arise as it can in poor areas of an urban area.
With respect to the west v. east aspect of this, I would be curious to know what your source for this is since I have seen statistics (as in this link) that show the West is not more violent than the East. However, if the West is more prone to crime, we might look at social stability. We might surmise that the East has been settled longer and is in less of a state of flux. It therefore has a more stable society. In the West, with lots of immigrants and migrants from other reasons, there is less stability and crime can be more prevalent.