or did I get something wrong about history?

I'm afraid that this is the case; the Medieval Water Thing is much overstated in popular culture, as the VFAQ explains (Middle Ages, then Health & Hygiene, in case your browser doesn't take you directly there). You'll want the first two answers, by u/sunagainstgold and u/Qweniden. As you've expressed the Muslim Variation, an answer from u/CptBuck here addressing how desert civilisations do for water is also in order.