Can science even investigate the supernatural?
Yes, of course science can investigate the supernatural.
The thing is that when science investigate and explains the supernatural, it calls the result "natural." So the effect is to diminish what is considered supernatural, and to increase what is considered natural.
Much that was once in the province of supernatural religion is now natural. If the religions were smart, they would embrace the natural and declare that their God is nature itself. However, they instead continue to insist on a supernatural God, and their God is being diminished as science moves areas from the mysterious supernatural to the understood natural.
Fundamentalism - the anti-American, anti-Christian branch of American Christianity