(please tell me at what point in the game this turns around...is it a technology, a building, moving to a different age, or what)
It turns around when you reduce your costs and increase your beakers or gold output. This means building commerce improvements with your workers, building courthouses, libraries, markets, just growing the cities out, etc.
You should never let your slider drop to 0% and get your units disbanded, though. You can expand quickly without that happening if you emphasize commerce in your cities. Also if you're going for a quick expansion strategy (without war) then you should make sure you aren't paying anything in unit costs. You can't afford to pay both extreme city maintenance and unit costs at the same time. Also, don't use high upkeep civics like organized religion, vassalage, or even hereditary rule (only medium I know).
But generally speaking I wouldn't advise a fast expansion strategy. It's best to find a balance. With enough games you'll figure out how quickly you should expand on a given map with a given strategy and then when to go to war to expand again. It sounds like you did much better in the second game.