Actually Australia is smaller than the US (Australia 7,617,930 sq km, USA 9,161,923 sq km). Australia is roughly the same size at the lower 48 states.
What Australia lacks isn't space (we got loads of it), it's water. Australia is the world’s second-driest continent (after Antarctica), with average annual rainfall below 600 millimetres (mm) per year over 80% of the continent, and below 300 mm over 50%.
As a result Australia's population density looks like this (1 dot = 1000 people)