Check out the debate between Deane Galbraith (cf. the podcasts he collected here), James McGrath, and Bill Heroman. Interestingly, I came to a similar position as Bill in this post where I suggested that is possible to reconcile the stories (e.g. Luke has Joseph transport Mary from her hometown in Nazareth to his temporary lodgings in Bethlehem where Jesus is born and shepherds visit, while Matthew has Joseph and Mary staying in a more permanent home in Bethlehem with a Jesus who could be almost 2 years old when magi visit), but that the major obstacle is Luke 2:39 and its assumption that Mary and Joseph returned to Nazareth as soon as they completed the prescribed temple rites. I looked at this issue from the perspective of the Synoptic Problem and whether Luke did not know Matthew’s account (i.e. did not know about a return to Bethlehem and flight to Egypt before going to Nazareth) or knew it yet purposely chose to skip over it. The theological emphases in each individual Gospel should also be stressed: Matthew’s Jesus is a new Moses who escapes a tyrant’s plot against the infant boys and comes out of Egypt and Luke’s Jesus is set in a particular imperial context (e.g. the census) and reaches out to the poor and marginalized (e.g. the song of Mary, the shepherds).
Update: Deane Galbraith critiques harmonizing interpretations here.
Update II: Jonathan Bernier nuances the discussion here by distinguishing between contrasts and contradictions.
Update III: Neil Godfrey criticizes scholars here for being more concerned about the details of narratives set in Bethlehem 2000 years ago than for the contemporary political situation.
The Jesus Memoirs 