After Arda became round at the destruction of Numenor and the removal of Valinor. The New Lands appear on the other side of the world.
In the sense that Arda is roughly analogous to the real world, the New Lands are thought of a The Americas.
It is often speculated that the New Lands was previously The Land of The Sun, where the Sun rested during the night when Arda was flat. I think that is valid but I want to add something.
When Valinor was removed I don't think the actual continent, Aman, was fully removed. It's mystical unmarred quality was removed as well as the Aniur and Elves who lived there, and it's cities and buildings. But I think the land itself remained.
I think when Arda became round the Land of the Sun in the East and Aman in the west came together to become the new lands. With Aman roughly analogues to North America and the Land of The Sun roughly analogous to South America.
And The Dark Land would be Australia.