I might have missed the explanation while reading the visual novel but if the magic and all that kinda stuff are kept a secret from the public, then how exactly do you explain the huge glaring flashes of light and the huge amount of noise that comes from the fight between the servants? Is it something to do with the Magic Association/ The Church? I don't actually remember any instances in the the visual novel other than the start of the series with Shirou getting stabbed by Lancer where any witnesses were killed. Not to mention the other shenanigans like Rider summoning his chariot.
From the author Kinoko Nasu himself:
The Holy Grail War is unknown to the general population, but to what extent is it hidden from normal people? Are the Church people (supervisors) manipulating the media and police (or even the governmental level) to cover it up?
A: The Church manipulates information, while the Magic Association handles the media. Also, the Association has the Einzbern cover the cost of any collateral damage.
Eyewitnesses can be dealt with by manipulating their memories.