I have a bathroom without an extractor. If I don't open the window the mirror fogs up in seconds of turning the shower on, and over time I get mould. If I do open the window it stays clear the entire time. By the time I'm out and dry the moist air has left and I close the window.

This morning it was -5C, so not cold by continental standards, but certainly cold enough.

In the UK recirculating air is very rare. In cases that bathrooms don't have a window (or indeed in new houses where they do) there's an extractor fan, but that just vents the air directly outside.