90 Honda Civic Idles fine, overheats as soon as you drive

Engine Cooling problem
1990 Honda Civic 4 cyl Front Wheel Drive Manual

Hi everyone, the other day I was driving my car and it was a pretty hot outside. I had the a/c on and while driving around town I suddenly noticed the temp gauge near the high point. I quickly shut the car off and let if cool. I open the radiator cap and noticed it was a bit low on water so I filled up the radiator.

Anyway, I was near my house and I drove it home without the temp getting passed the midpoint. When I got home the first thing I did was change the thermostat, it hadn't been changed in a while. I then turned the car on, let it idle for a few mins and the temp was fine. I then drove it around the block. The car quickly started getting hot. In about 3 mins the car was reading 3/4 up on the temp gauge.

So now I thought the radiator was clogged because inside the radiator was somewhat rusted and the upper radiator hose was hot and the lower was cool to touch. I then put in some radiator flush solvent and followed the directions. After I was done with that I let the car idle for ten mins and the temp was fine. I then drove no more than 1000ft and the temp began quickly rising. I then popped the hood and noticed water/coolant being forced out the reservoir and the upper radiator hose hot and the lower cool.

I'm assuming the radiator isn't clogged because I flushed it twice and I even put the high-pressure water hose in where the cap sits and water was easily able to run out the radiator's draincock.

At this point I'm thinking its the waterpump. Anyone know exactly what the problem is?
Sorry for the long post but I wanted to make it as detailed as possible so that I do not take up too much of your time. Thanks.

P.S. I'm certain the thermostat is installed correctly.