Hate to be the bearer of bad news, but shit is literally aerosolized any time you flush the toilet. And it’s not contained the bathroom. And it doesn’t matter if the toilet seat lid is up or down.

Mythbusters did an episode on exactly that. It is worse than you’d think. I can’t find the actual episode right now, but someone wrote an article about it/the findings.

The other brushes were placed elsewhere in the home, including the kitchen and even an office on the other end of the building, and all of the other ones were rinsed daily but not used for brushing. At the end of the month-long experiment, the toothbrushes were analyzed by a microbiologist, and they found that every toothbrush had a microscopic amount of fecal matter on them, regardless of the distance from the bathroom. source

Bidet or not doesn’t matter. Shit is literally all over EVERYTHING. ALL the time.