1999 Ford Taurus ALT

Electrical problem
1999 Ford Taurus V6 Front Wheel Drive Automatic 81,000 miles

I was driving my lights were flickering then I notice that it looked like their was a short or something. I checked the wires they were ok. We were on the road again and then I noticed that the batt light was on dim and flickering as we drove along the batt light got brighter lights got dimmer well we pulled over and turned everything off to recharge the batt and it did so I thought so after a short time it died the car batt/the problem is I put a new alt in because it tested bad, charged the batt started the car and it ran for 15/20 minutes? Alt is ok battery is ok but know charge gets to battery? Where to look