I used to have a bathroom without an extractor and in the winter with the window open the cold air would just cool the walls and cause more condensation there than I’d get with it closed.
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.
