In addition to Matthew (possibly 55-70 AD, with surviving sections dated to 50-99 AD), Mark (possibly 55-60 AD, with possible sections found in a mummy's mask from 80-90 AD), Luke (possibly 55-63 AD, with sections dated at 175-225 AD), John (possibly 80-90 AD, with a surviving fragment from 125 to 135 AD), and Acts (possibly 60-70 AD, with fragments from about 350 AD), all naming Nazareth (even in the Greek), Justin Martyr [110-165 AD] wrote in Chapter LXXVIII of his "Dialogue with Trypho" (written in about 155-165 AD, considered "undoubtedly genuine" even by critics) that:
And Joseph, the spouse of Mary . . . went up from Nazareth, where he lived, to Bethlehem, to which he belonged . . .
The idea of writing at a time when people could and did visit Nazareth and claiming dishonestly that the town had existed a mere 50 or 150 years earlier, without the lie being discovered, is absurd. Anyone led astray by such absurd claims wants to be deceived - a category that includes an unfortunately large fraction of today's population.
Anyone who thinks Nazareth didn't exist in 4 BC probably reads this quote from Lincoln (taken from his daily blog entry just after the Gettysburg Address) and misses the humor. It's like claiming the My Lai massacre or the Battle of Gettysburg could not have happened because the towns didn't exist in 1968 or in 1863 - we know they existed 50 and 150 years ago. Most people have a good idea what was happening 50 or even 150 years ago, at least in high-profile areas, and Nazareth was high profile once the Gospels were circulated.