Being terrified of death, although we can logically reason against being so, is perfectly natural. Our species wouldn't have come this far and evolved so greatly if we didn't have some predisposition against death.

Almost every thinking creature today fears death, because we have all evolved to fear death (avoid death might be a more accurate description in the case of some lesser intelligent species). Any genetic line that doesn't fear death is much more likely to die out.

We, as humans, are unique, because our reasoning skills are to the point that many, though not most, can overcome this primal fear.

Even so, when faced with imminent death in the course of daily life, practically every person in the world will fear. Even if we have consciously rejected fear of death, when the gun is in our face, when the train is barreling towards us, we will feel fear. Because though we are highly intelligent, when the shit really hits the fan our conscious minds freeze, and we are directed by our instincts. And our instincts will always fear death. Because if we had not evolved that way, we would have died out long ago.