Was baby Jesus in two places at once?

The stories of the birth of Jesus Christ, recorded only in the gospels of Matthew and Luke, are mutually exclusive in their details.

Matthew states that the magi visited Jesus in Bethlehem shortly after his birth, but after being warned in a dream, they did not return to Herod, who had sent them. Joseph, also warned in a dream, immediately left with Mary and Jesus for Egypt until Herod was dead.

Luke states that after the birth, Joseph and Mary took Jesus to the temple at Jerusalem to present him to the Lord.

The Jerusalem visit could only have fit in with the Matthew account if it happened before the visit of the magi to Bethlehem, because once the magi left Bethlehem, Jesus' family left immediately for Egypt, according to Matthew.

But Jesus' family could not have gone back to Bethlehem after the visit to Jerusalem, according to Luke, because he states that they returned to Nazareth, their hometown, not Bethlehem.

Further, if the magi's visit to Bethlehem occurred after the family went to Jerusalem, where was the guiding star of the magi pointing in the meantime?

It appears that the accounts are mutually exclusive and that therefore one or both of them are wrong and that therefore one or both are not inspired by God. Do you agree with this assessment?