Where does the New Testament clarify that the "Word" in John 1:1 is referring to Jesus?
That is a good question. On thing that makes it so puzzling is that John tends toward long sentences sometimes, and then goes back to the original thought.
We know of whom John was speaking when he talked about the Word if we just go a little farther in that same chapter (John 1). The first thirteen verses are essentially one long sentence. When we go to the next sentence (verses 14-17) we see who “the Word” is. “And the Word was made flesh and dwelt among us, (and we beheld his glory, the glory as of the only begotten of the Father,) full of grace and truth. John bore witness of him, and cried, saying, This was he of whom I spake, He that cometh after me is preferred before me: for he was before me. And of his fulness have all we received, and grace for grace; for the law was given by Moses, but grace and truth came by Jesus Christ.”
If there is any doubt that he is saying that Jesus is “the Word,” then we only have to see of whom John spoke when he said “He that cometh after me is preferred before me.” To find that out, we only need to go to verses 29-30. “The next day John seeth Jesus coming unto him, and saith, Behold the Lamb of God, which taketh away the sin of the world. This is he of whom I said, after me cometh a man which is preferred before me: for he was before me.”
So the progression is: In the beginning was the word; John said of him that he who comes after is preferred before me; John saw Jesus and said he was the one of whom that was said. Therefore Jesus is the one John called the Word.