It’s called toilet plume.
A toilet plume is the invisible to the naked eye, cloud-like dispersal of raw, infectious sewage particles as a result of flushing a toilet. Flush particles rapidly rise out of the bowl and several feet into the air after flushing. Aerosolized droplets can remain suspended in the air for tens of minutes between users, posing an inhalation risk. These invisible particles go on to coat and spread onto surfaces like floors and counters or objects like hand towels, bathmats, or even toothbrushes.
I wouldn’t consider being cautious about having literal fecal matter in my toothbrush mundane, but you do you.
Everything everywhere is covered in fecal matter. Mythbusters did an episode on it and their control toothbrush in the office had plenty.
Well you should be far more worried then. Because closing the lid or not won’t matter. Your toothbrush is covered in it anyway. Unless you’re covering your toothbrush when you’re not using it.
I keep it behind the cabinet mirror, I’m sure over time that’s better than in a cup next to the shitter. If you’re comfortable keeping your toothbrush out and near the toilet than more power to you.