How can economic growth reduce carbon emissions?

There is an argument I can’t get my head around. As economies grow, they adopt cleaner energy. But does the aggregate growth of output—even with the use of cleaner energy—still result in less overall pollution than if the economy hadn’t grown in the first place?

For example, I live in Canada. We have a lot of hydro and nuclear power. But our carbon footprint per capita is larger than the United States, more than twice that of China’s, and many times more than less developed countries, even though they use dirtier energy. I know the USA’s carbon emissions have marginally declined recently because of things like fracking. And I realize that developed countries like France, Denmark, and Sweden have 13 of our carbon footprint. But we are a cold country with sprawling cities. Can less developed countries reach our GDP per capita while still maintaining low aggregate and per capita carbon footprints without some game-changing technological advances? Can we get to less carbon emissions than France or Denmark without rebuilding our cities?