Here’s a closer look at whether or not there is a contradiction:

When dealing with sceptics’ claim of Bible contradictions it seems one can never be reminded enough of what exactly is a contradiction. A contradiction occurs when two or more claims conflict with one another so that they cannot simultaneously be true in the same sense and at the same time.
Technically the sceptics’ premise “They went to Egypt after Jesus’s birth” and “They went to Nazareth after Jesus’s birth” are not necessarily contradictory. Joseph and Mary could have went to Egypt and Nazareth at different instances after Jesus’ birth. For example, Joseph and Mary could have went to Egypt first and then later Nazareth or vice versa, they went back to Nazareth and then Egypt. But notice the discussion of which place they went first is no longer an issue of contradiction but chronological order.
We must not make the chapter in Matthew and Luke to be a greater conflict than it ought to be. Looking at Matthew 2 and Luke 2 we see that both are in agreement that Joseph, Mary and Jesus eventually settled in Nazareth. This is seen in Matthew 2:23 and Luke 2:39.
If Joseph and Mary went to Egypt first before going to Nazareth, we still do not have a contradiction between Matthew 2:14 and Luke 2:39.
The verb for “returned” in “they returned to Galilee” in Luke 2:39 is ἐπέσ”ρεψαν. It is in the aorist tense which typically convey the idea of the action been seen as a whole. That is, Luke 2:39 can be understood as talking about Joseph and Mary returning to Galilee in general, and not the manner and other aspects of how they returned. Thus, Luke 2:39 was never meant to speak beyond the action as a whole of their return to Galilee and can’t be used to contradict the details in Matthew 2:14.
From the point above we also must remember that silence in Luke 2:39 about Joseph and Mary going to Egypt must not be taken to mean the same thing as a denial of Matthew 2:14 that they went to Egypt. Again, here we do not see a contradiction.
Or if Joseph and Mary went to Nazareth first before going to Egypt and then finally back at Nazareth, we still do not have a contradiction between Matthew 2:14 and Luke 2:39.
Remember the Jews of Galilee often travelled back and forth into Judea. Luke 2:41 even record the fact that Jesus’ parents went every year to Jerusalem and apparently others also made such trips as hinted by Luke 2:44 with the mention of “caravans.”
Given how regular and routine Galileans travelled to Judea and back, it would not be surprising that after the wise men found Jesus at Bethlehem (Matthew 2:5-8) that Joseph and Mary went back home after the census. For a new married couple, they might not have all that much resources to stay that long in Bethlehem more than they need to and thus went home.
Why would they flee from Herod if they were already in Nazareth and Matthew 2 record that Herod killed the children around Bethlehem and in Judea? Don’t forget they just had taken a census and no doubt Herod could have used that information to track down Jesus back to Galilee. Galilee was also under Herod’s control, in fact Galilee was the first area Herod was in charge of when he first was given power. So it makes sense to flee to Egypt even if they were already in Nazareth.
Again, there is not a contradiction.