That's why toilets flush - to carry everything away. If it's not working right and cleaning properly, it sounds like the toilet is defective.
Ask the hotel staff to:
- Fix the toilet
- Or move you to a room with a better toilet (upgrade opportunity)
- Or to provide a toilet brush (if you prefer)
If you're just stuck with the poorly performing toilet then lining the bowl with toilet paper before using it (even the "dry" side parts or wherever the problem areas are) should help.
Or use some more toilet paper and something disposable like a straw or rolled up paper to wipe it off, then throw everything in the trash.
Or maybe you're not using the right "button" on a dual-flush or electronic toilet, there could be a "heavy flush" setting you're missing.
Another idea, if your hotel has a hand-held or detachable shower head on a hose, and it's right next to the toilet you could
Or, you could make a "DIY Personal Bidet" by poking a hole (with a pen/key/fork/etc) in the lid of a cheap water bottle, so when you squeeze it water sprays out, and use it to spray clean the toilet & anything else.
I've never heard of a toilet brush being used as "permanent toilet paper" that sits in the bathroom, to wipe up feces without any cleaner, that sounds like a biohazard. You probably shouldn't use a toilet brush without some type of cleaner / soap.
You can actually buy a Handheld Bidet Toilet Sprayer that attaches to your toilet, very similar to a kitchen faucet sprayer. I would recommend using one, instead of leaving poop on your toilet brush all the time ;-)
[ If I found a house guest had put feces on the toilet brush and left it there, I would not want that guest back in my house again, ever (and I wouldn't touch or eat or drink anything from their house - or maybe I'm a little paranoid about Hepatitis, Cholera, Salmonella, E-coli, Enterovirus, Norovirus and parasites...) ]