Harry didn't ride in the thestral-drawn coaches his second year. He and Ron arrived in the bewitched car. His third year he did ride in the coaches but didn't see the thestrals.
At the end of his first year though, he killed Professor Quirrel with his bare hands. Why couldn't he see them before Year 5, after seeing Cedric die?
It says: "Harry does not see him die in the book, although he does in the film. This error will cause some problems later in the story, since Harry should be seeing the thestrals already in his third year if he had seen Quirrel die."
So according to the book, he didn't see Quirrel die, but he saw Cedric die at the end of book 4 and sees the thestrals at the beginning of #5.
I so wish the movies would stick to the facts!! It gets too confusing!!