Whatever you're reading is perhaps confusingly put. 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').

So that's a fringe idea. A much more mainstream idea is that, not the actual wording of the Iliad, but the tradition of formulaic language used to compose it, goes back many centuries before the Iliad was composed. By 'formulaic language' I mean the typical phraseology that appears again and again, and which some translators try to conceal: things like the typical phrases 'Agamemnon king of men', or 'he fell with a crash, and his arms clattered upon him', or 'so he spoke, and in answer so-and-so replied': things like that.

That has the aura of plausibility, since some formulaic language looks very archaic, and it's widely taken as gospel particularly among American scholars. Then again, one of the reasons it looks archaic is because lots of it comes from relatively conservative dialects of Greek, especially the Arcado-Cypriot dialect, but also Aeolic. It has been argued by the linguist Dag Haug that the formulaic system isn't nearly as archaic as once thought. That's an ongoing area of investigation.

Anyway, I'm rambling a bit. The main point is that the Iliad isn't a 13th/12th century composition by any stretch of the imagination. It dates to the first half of the 7th century. It may still have taken another century after that before it got written down, but if so there'd be a good reason for that, as I described above.