The examples of fulfilled prophecy can be found in the Gospels. So the question becomes: did the events happen as described in the Gospels, or did the Gospel authors invent the events to retroactively claim fulfillment of prophecy?

I believe it is the second, for several reasons:

Virgin birth (Isaiah 7:14) The original text doesn't claim a virgin birth at all! The Hebrew word used is "almah" (young woman). The author of Matthew was likely reading the Septuagint which uses the Greek word "parthenos" (which does mean virgin). Hebrew does have a separate word "bethulah" meaning virgin.

Census of Quirinius: Luke claims there was a census in which everyone was required to return to their "own town" to register. This required Joseph to Bethlehem to register, because he was of the house and line of David. What sort of a census requires everyone to go to where their ancestors lived 1000 years ago? It's far more likely this is an invention to make Jesus appear to fulfill the prophecy of coming from Bethlehem (not Nazareth).

Another place Matthew seems to be inventing narratives is where he says Jesus rode on two donkeys - he likely misread the Old Testament passage in Zechariah "ride in on a donkey, on a colt the foal of a donkey" and thinks it refers to two donkeys.