first of all: Happy New Year!
Depends how clean you like it.
Mostly I am willing to spend 50 EuroCENT (not EURO only Cent!!!) for a clean toilet.
Good places are railways stations, department stores, self-service restaurants, hotel lobby of major hotels. I try to avoid any public toilets without attendants.
The more south you are in Europe the more acceptable it is just to ask to go to a restaurant without consumption.
In some Eastern European countries I would recommend to have some adults diapers and just use them in a case of emergency.
Mostly you find signs like "WC", "00", "PP" (only men), or pictogram of man and woman.
Do not worry: the majority of Europeans know the use of toilets and appreciate them more than to pee in the booshes or to trees.
I am sure you will even have less problems than you might think. Especially South Europe is very civilised as private bathrooms have Bidets and in Turkey all toilets have a water tube below the seats which allow you to clean your ... very carefully.