The Old World is a reference to those parts of Earth known to Europeans before the voyages of Christopher Columbus; it includes Europe, Asia and Africa.
From the viewpoint of those early explorers I suspect that North America was little different to where they had left. By that time they almost certainly had many generations of experience of travelling, on and along, the ice sheet and one end of it was pretty much the same as the other.
It certainly makes a lot more sense than the southern Atlantic route.