Car started fine, but the alternator was not charging it, battery drained again

Hi, I'll try and give a series of events as accurate as possible.

One day the car wouldn't start. I figured I left the lights on or some other mistake.
I bought a new battery and installed it.
Car started fine, but the alternator wasn't charging it, battery died again.
Bought a new alternator and installed it.
Boosted the vehicle, it started.
Car shut off after about 5 minutes of running (presuming it was still running off the battery).
Checked the alternator fuse with a multimeter because I'm out of ideas, fuse seemed fine.