I think that this is canonically untrue. Federation vessels do have multiple bathrooms or "heads" in military jargon, including personal ones attached to officer quarters.

However, Memory Alpha does corroborate the version I remember, which is Frakes talking on some behind the scenes special about how there's only one bathroom on the entire Enterprise-D. Scroll down a bit to the "Background" section. https://memory-alpha.fandom.com/wiki/Bathroom

(And if you're surprised there's a MA page for just this topic, I am too, and probably shouldn't be.)