Battery isn't charging

you didnt check for a fusable link on the wire to the altInater??

Sounds like the fuseable link for that alternator circuit or the main power fuse for the fuse box is open.
Make model and year ?

Sounds like you fried a circuit or two. The battery light means the alternator is not charging the battery which you know.
The charging circuits will need to be checked, it could be a fuse or a fuseable link which is a wire that acts like a fuse.

If you pulled the fuse for the ignition you killed the coils and module.
I assume you had the battery tested for a bad cell.
There are fuseable links in the electrical system in addition to the fuses you can see. Everything has to be disconnected when testing for a power drain. It can be very difficult to find on today's vehicles.

ok, first, once the vehicle is started, the alternator should take over the functions of the car,, not the battery , the battery should charge during this time, ( car running) if the alternator is having to supply to much juice to the battery the whole car suffers, and the battery never fully charges . Just because the battery says 12 volts, doesn't mean the amps are there, which you need to start the vehicle,, take the battery and get it LOAD tested, this will let you know if one or more cells in the battery aren't up to par, if it doesn't measure up,, get it replaced, you sate that its only 4 months old, so take it to someone other than who you bought the battery from, and have it load tested there, once you have a known GOOD battery ,, start the vehicle,, then with a meter check the battery posts,, you should get a reading of between 13.5-14.5 volts, if you don't,, your alternator isn't doing its job and will need replacing

The altinator is missing ground, any time you remove the battery and the engine dies, that only means that the altinator is not charging, there has to be a fuse for charging systme.

Sounds like a later model with computer controlling the output and this is faulty ,now one of the wires on the back is for the warning light in the dash so disconnect one of these thin wires at a time to locate the warning light control ,then refit the warning light wire and disconnect the other other one ,now what you do is to run another wire from this terminal to a 12v ignition switched feed in teh fuse box wrap wire around leg refit fuse and with engine running disconnect battery and it should keep running if its working ok as the alternator is now battery sensed and not computer controlled ,easy just like an older car without all this fancy electronics which can and does go wrong

Could be..... the batery is just used for starting, once the engine is running it pulls electric power from the altinator, if the altinator which also charges the batery, is not working power is pulled from the battery, and when the battery goes dead, the car does too. When the Car turns off, do you then need to jump it to start it? if so you could be pretty convinced that the altinator is the issue

Change voltage regulator. Does the battery take a charge from a charger? If not, replace battery. If it does, then regulator is faulty. If still nothing, return alternator to place of purchase, and have it checked.

I would check with vom voltage at battery terminals and at starter to ground. Is possible could be a fuse or fusable link but I think the positive goes straight to starter and power distribution fuse or fusable link