The fan is attached to the water pump hub, so I would totally expect it to run when the engine does. It just doesn't run as fast when it's "cold" as it slips. When it hits the temperature point the clutch locks up and pretty much is 1:1.the fan runs all the time, is that normal if it’s a thermal clutch ?
If it never locks up, it will slip and not move as much air as it should when it needs to, thus can not help you control temps.
If it goes the other way and locks up all the time, something is wrong with it and while it won't hurt your temperatures, it has failed and needs replacement. Will make a lot of noise and is totally annoying. That's not as likely, but can happen.