Closing the game, whether that's clicking the X on steam, or navigating away in a browser, will pause a moment and prompt the game to save immediately. I suspect that closing the browser completely won't allow this, so I would close the Crusaders tab before closing the browser, if possible.
I'll dig more into the limits that various browsers might impose on saving in this way.
And in general, most meaningful actions (moving Crusaders, leveling them, buying upgrades, beating areas, bunch of others) will prompt the game to save within a couple minutes. I have some ideas to have it save more aggressively in these situations without spamming the server as it did previously.
For now, if the saving on close isn't working for you, I'd move a Crusader around and wait for the "Saved!" popup. Sorry for rambling!