86 pontiac grand Overheating problem

A few weeks ago I brought 86 pontiac grand am 4 cyl. Is overheating. Electric fan wasn't working, Replaced with used one, have connected to battery so it runs all the time key is on. Still overheating. Water pump is working, I can tell by feeling top hose. At start up water flows though hose slowly, but gets up to 220 it flows good. Thermostat has been taken out before I got it. (now I have heard some cars must have one in) How true is that? Have sprayed water around cly head and no changes it RPMS. (was told by doing this I could tell if it sucking air) There are no leaks any where, no water signs. Anti-frezze is ok. Let car get to 220 and shut it off, water will steam into over flow tank. Never seems to be loss of water. Of course I shut it off as quickly as possible, but ifr it gets to 260 or above ( highest gauge goes) I let it cool, check water might be pint or two low. I am lost and could use some help. Could it be the radiator, but water flows though good. Please give me some ideals. No water in oil, I see no air bubbles. Tank you all. Yes it needs a thermostat to maintain a good operating temp. If you run without one (I know this from experience w/our firetrucks at the station) you can heat all the coolant up and when the engine is running at high rpm the water will not have the suffiecent time to cool in the radiator, causeing an overheat.

I'm not sure how spraying water on the heads or intake would change idle speed. I have always done this with starter fluid or an unlit propane torch. Whatever floats you boat though. I will get one in right away and try that. Thank you. If any one else has any thoughts I would love to hear them. Thank you all Now put thermostat in, drove car around 20-25 mintues. Never got above 220. Before it would of reached 260. But stop turn engine off water fills the over flow tank with full force, and water is hot. Cools water goes back into radiator. Is this Normal or what do/can I do next? Thank You Have you tested or replaced the radiator cap?