Link does fight Link but i don't describe itLink in TP originally fell to the power of the Fused Shadows, as the cut scene portrays. Since he fell prey to it, he became dark, like the dark interlopers in form. The goddesses (or the Light Spirits, can't remember which one right now) came to seal him away like the dark interlopers, but right before they did, he used the fused shadows to travel back in time.
He contemplated for a while, and began to hate his past for falling to the Fused Shadows. He tried to destroy himself, but found he couldn't. He suddenly thought that if he destroyed his past, he himself would be destroyed. So from then on he began time traveling to find links to him so he could destroy it.
This is how we come to see him in Oot. He wanted to destroy Link, and so did Ganondorf, so Ganondorf enlisted Dark Link to kill Link in the Water Temple. Link defeated Dark Link, but he wasn't killed, so Dark Link had to look for other links to himself.
P.S. When Dark Link traveled back in time, he created a parallel universe, where Link didn't fall to the power of the Fused Shadows. __________________