The pistol is a glock 20 with 3dot sights.
I normally shoot 180gn at 1200 fps.
Today I took out some 150gn at roughly 1350 fps and it was shooting really low (1.5ft low at 5-10 yards) The only way I could hit anything was to line up the top of the rear site with the very bottom of the front site where it meets the frame.

Then, just to check, I went back to the 180's and right back on target.

Is it normal for a change in bullet weight to throw off the sites that much?

Yep. Bullet weight is the #1 factor in recoil. The bullets hitting higher is because of increased recoil which begins before the bullet leaves the barrel.

it's more common than most think. The heavier load will shoot higher due to more recoil and slower bullet.

the recoil on the lighter faster bullet was significantly more than the heavier slower one but the heavier recoiling, faster one is the low shooting one. Is his because the faster one leaves the muzzle before it gets a chance to rise as much?

Either way I think I'm going to be sticking to 180-200gn bullets from now on so I can actually hit my target.

I've noticed this even in a 40 S&W

That has been my experience. Slower bullets, even with greatly reduced recoil, leave the muzzle later in the "rise curve" so are pointed higher at the target. Most pistols with fixed sights are sighted for one particular weight.

I had an SP101 that would put 158 grain bullets right at POA, but would put 125 grain bullets a good 18" low at 15 yards since they were going so much faster, i.e. leaving the muzzle so much sooner.

My Flattop .44 Special sighted in with 200 grain bullets at 950 fps puts that same bullet about 6" higher using Trail Boss at 650-700 fps.

The difference in bullet speed plays a smaller role. The biggest factor is recoil. The actual physical push is more with the heavier bullet. The lighter load often "feels" or "seems" like the heavier recoiling because of the increased muzzle blast & associated noise. Feel, sight, & sound; they all come into play in how we perceive recoil.

The only way to really understand that the lighter bullet that "feels" like it's recoiling more is to shoot rapid fire against a timer. Then you'll see that the supposedly "hotter" light bullet load actually is generating less recoil impulse, allowing you to get back on target quicker. I had to see this myself back when I shot IPSC and went from a non-comp .45 ACP to a compensated .38 Super. The Super felt more "whippy" and it was MUCH louder due to the comp. But my split times took a huge drop; I was shooting much faster even though it all "felt" like there was more recoil. Things aren't always what they seem to be. Don't take my word for it, take the ballistic data from the two loads and input it into an online ballistic calculator to calculate the free recoil.

The reason your lighter bullet - traveling at a faster velocity impacts low is TOTALLY RELATIVE to your original zero with the heavier, slower bullet.

During the impulse of recoil the lighter, faster bullet spends less time in the barrel and generates less relative recoil than the heavier slower bullet - so it impacts low relative to your initial zero.

As the others have mentioned, this is very common. I notice it most if I switch from 180 to 240 or 300 grain .44 mag loads out of a 6.5" barrel.