Why do you think that the fan relay is bad? Relays are dirt simple to test, and they pretty much work or do not work. Check continuity/resistance. Apply power across coil, check continuity/resistance. Did it close the circuit? If so, it works. If not, it doesn't. Point being that ordering a replacement because you
think it's bad isn't a good idea.
I will assume that you don't know how to "burp" your cooling system.
Jack up the front end, which will ensure that your radiator cap is the highest point in the system. (This is different from jacking up your radiator cap
![Stick Out Tongue :p :p](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
) May not be 100% necessary, since we're going to force the system to circulate, and the air will find its way there anyhow, so maybe it just makes me feel better to do this. (I usually do this while the front is on ramps.)
When the engine is cold, remove the radiator cap. Hot engine + open radiator cap = visit to the local burn unit, and not in the candy stripper charity way.
Start engine. Turn heat on full (Or at least off full cold with the automatic A/C unit) to open the heater valve, and make sure that the water pump is circulating through the heater core. Let the engine warm a bit. Watch for signs that coolant is moving through the radiator (IE: thermostat open.)
Bring up the RPM to 1.5-2k. This makes sure that the water pump is working better than it does at idle, enough to maybe force some air bubbles to move through the system. Eventually said bubbles will find their way to the water neck, through the open thermostat, and into the top tank of the radiator, where you will shortly replace it with fresh coolant.
My experience is that there will be a few bubbles, Probably you'll only see a very slight drop in fluid level at the cap, often preceded by rising levels at the cap while the engine warms. During this period some coolant spilling from the radiator cap is normal. Air will expand a lot more than the coolant will as it gets hot. After a bit the coolant level will go back down. Don't expect to see bubbles in the coolant, the fluid level will just go back down a bit. Having the front up high makes it easier to catch the dripping coolant before it ends up on the driveway, or at least I find it so.
Fill the radiator. Fill the overflow coolant bottle to near the "full hot" line.
Test your radiator cap. If you can't test it, and it's more than 2 years old, just replace it. They don't really last particularly well. I don't test mine anymore, I just replace them when I do my coolant flushes. They're cheap enough to be treated as disposable items.