Claim: the Iliad is usually dated to 13th to 12th century BC and written down ~8th c. BC.
Response: A speculative historical Trojan War is customarily imagined as taking place in the 13th or 12th century BCE, on the supposition that it happened at all. But the Iliad itself is very definitely a 7th century poem. Writing wasn’t used in the Greek world to write down substantial poetic/literary texts until at least that time, and probably not until the second half of the 6th century. Some epigrams do appear before then, in the 8th and 7th centuries, but never anything more than a handful of lines. Linear B fell out of use around 1200 BCE, and it doesn’t appear to have been used for literary or poetic texts of any kind. So these are the main reasons we don’t have any Bronze Age Greek poetry.
With the Iliad, it was long suspected that while the poem itself isn’t Bronze Age, it uses Bronze Age material. In terms of content, storyline, characters, and so on, there’s no way of conclusively settling that one way or the other — not to the satisfaction of someone who believes the opposite, anyway. In terms of poetry, we can be extremely confident that there isn’t much actual poetry, as in chunks of words, that dates to the Bronze Age or even much earlier than the 600s BCE.

There are some older scholars — a tiny minority — who believe that some chunks of the Iliad did take their essential form in the Bronze Age, such as Richard Hope Simpson, who wrote a book in 2018 arguing that the Iliad’s Catalogue of Ships (in Iliad book 2) is essentially a Mycenaean poem. But they are a tiny minority, and the idea is entirely unsustainable.
To give one straightforward example, the standard word for ‘and’, καί, is post-Mycenaean and its usage indicates a conflation of at least two dialectal forms derived from a Bronze Age preposition *kati ‘in conjunction with’. But it is one of the most frequently occurring words in Homer. It appears very heavily in the Catalogue of Ships, for example (as does a late dialectal form of the word for ‘ships’).