Is it true Women's bathrooms are worse than Men's?

Men seem to have this thing where they believe that women are mostly clean. This usually translates into people thinking that women's toilets tend to be cleaner than men's. I have heard otherwise. Which is worse and what are your methods to stay clean in a public bathroom? My grandmother on my dads side genuinely carried her own toilet paper for public bathrooms, do you keep paper or tissue on you?

I personally will avoid any public restroom. It's been years since I last used one as I always go before I leave the house to avoid it.