Well, yes and no.
No: According to the Quartz Hill School of Theology (I will give leeway to the obvious bias in the site, however the dates seem to align with other scholarly sites in my emphasis):
Though some scholars would claim a composite authorship for these books starting around 950 BC with them reaching their final form during the time of Ezra around 500 BC, I would take a more conservative position and argue for a unitary authorship within fifty years of the Exodus (which would date from either around 1290 BC or 1440 BC; that is another whole area of controversy), with much of the material coming from Moses himself
Ignoring the whole problem of there being no evidence for the exodus story, we can say that given a range of anywhere from 950BCE to 500BCE actually places the writing of the stories in the Iron Age.
However: These are the dates of the stories being written down! That means that the timing of these stories is such that they are a supposed narrative of events that took place as the author of the theological site claims, in the 1290BCE to 1440BCE time-frame. That actually places it in the Bronze Age. Now that is generally accepted as the books of Genesis, Exodus, Leviticus, Numbers, and Deuteronomy. Joshua is another ancient story, and as the theological site claims:
The book would thus date to either the 15th or 13th century BC, depending, again, upon exactly when the Exodus from Egypt occured (sic).
Again, ignoring the problem of the historicalness of the exodus... it is squarely in the Bronze Age.
If you don't want to accept what theologians who accept the default position of the bible as the ground truth, this page from the University of Minnesota gives the following dates for supposed authorship (again, note that this is when the books were authored, not the time-frame of the sorties they are telling):
B.C.E. Old Testament
c. 2166 to c. 1876 Job
c. 1446 to c. 1406 Genesis, Exodus, Leviticus, Numbers, Deuteronomy
c. 1406 to c. 1050 Joshua, Judges
With the remaining books being written later and later in history. Again, you see that some were written in the bronze age, while others were written in the iron age. Of course, the time-line of the events are somewhat argued, but for the sake of simplicity, we can go with Usher's time-line which places everything up to Exodus in the Bronze Age:
Ussher's chronology provides the following dates for key events in the Biblical history of the world:
4004 BC - Creation
2349-2348 BC - Noah's Flood
1921 BC - God's call to Abraham
1491 BC - The Exodus from Egypt
1012 BC - Founding of the Temple in Jerusalem
588 BC - Destruction of Jerusalem by Babylon and the beginning of the Babylonian Captivity
4 BC - Birth of Jesus
So, while the writing of the stories may not be bronze age, I think it is still a fair characterization based on when the events are alleged to have taken place.