Even when a thermostat is fully closed, even stuck closed, some fluid still circulates through the radiator...rough guess about 10%. When it's really cold or you have a really efficient system, this can very well be enough circulation to keep the engine at a perfectly normal temp with the thermo closed. Does happen. I have a F250 radiator in my truck and this time of the year especially on the freeway, I see engine temps hanging around 185*, and that's with a 195* thermo.
How are you verifying the water is not reaching a normal temp? Do you have an aftermarket gauge or is the stock gauge not reaching a position that you normally see?
So most likely, if it's not reaching temp for sure, then the thermo is stuck open. Even if the fan were cooling constantly, it would not fully overpower a normal working thermo and keep the engine too cool.
The fan should turn by hand when cold relatively easy, but you shouldn't be able to swing it around. I've never had one that would make a whole bother revolution on it's own.