Most toilets I've used don't fill the bowl with enough water on a single flush to overflow. You can adjust the chain (assuming not an office toilet) if it's flushing for too long.
The only times I've overflowed a toilet I was as a kid, not realizing that trying to flush again might make the problem worse.
I assume either they mean they don't keep flushing it over and over if the bowl fills up or their toilet flapper has a short chain and stops flushing quickly when they let go.
I was actually just reading just reading about this pretty recently. If I remember correctly, the manga has her explicitly possess a human shell when she needs to interact with humans, and she doesn't have those powers in that form. The anime glosses over that, so you just see her fly around a lot in the early episodes and then she stops.