[sideshow]

is your alt light an led
fk it off and run a normal old school 6w globe just for a test
if it goes out means yr alternator is sort of working fine until further tests

how have u wired the wires on the alternator
u might have not wired the s terminal porperly
L is the light whihc goes to neg side of the globe in dash
S is the supply which means battery or ign
on most alternators we just piggy back this off the big pos terminal on back of alternator
if your alternator had dropped a phase it would charge at 12v
so since its at 13v then its charging but i recon its not getting a correct reference from the S wire therefore not charging at capacity when u turn lights on

so make sure the s wire is getting good accurate voltage

otherwise take it to auto elect