Was baby Jesus’ life threatened in Jerusalem?
(a) Yes, so Joseph fled with him to Egypt and stayed there until Herod died (Matthew 2:13 23)
(b) No. The family fled nowhere. They calmly presented the child at the Jerusalem temple according to
the Jewish customs and returned to Galilee ( Luke 2:21-40)

The above is a claimed contradiction in the Bible accounts of Matthew and Luke. However, by combining the gospel accounts, we get a better picture of the events in Jesus’ life. 

“Eight days later when the baby was circumcised , he was named Jesus . . . Then it was time for their purification offering, as required by the law of Moses after the birth of a child; so his parents took him to Jerusalem to present him to the Lord”—Luke 2:21,22 NLT
In Luke’s account, Jesus’ parents took him to the Temple to be presented to God after he was circumcised on the 8th day after his birth. 

“Jesus was at Bethlehem in Judea during the reign of Herod. After his birth astrologers from the east arrived in Jerusalem . . . Herod . . . sent them to Bethlehem . . . and, entering the house, they saw the child with Mary his mother bowed in homage to him”—Matthew 2:1-11 REB

Matthew’s account picks up the events when the astrologers from the faraway east visited, which was apparently some  months after Jesus’ birth, when his family by that time living in a house. By this time Jesus was no longer referred to as a “baby”, but as a “child” (Matthew 2:1-12).

“Then they returned to own country by another route, for they had been warned in a dream not to go back to Herod”—Matthew 2:12 REB

The astrologers left Bethlehem and did not return to Jerusalem and King Herod. 

After the astrologers  left, God’s angel appeared in a dream to Joseph and told him to move the family from Bethlehem to Egypt to avoid Jesus getting killed by murderous King Herod (Matthew 2:13-15).

“When Herod realized the astrologers had tricked him, he flew into a rage, and gave orders for the massacre of all the boys two years old or under, in Bethlehem and throughout the whole district, in accordance with the time he had ascertained from the astrologers”—Matthew 2:16 REB

By carefully reading and analyzing what the Bible actually reports, the family’s 7 mile trip from Bethlehem to Jerusalem, for Jesus’ presentation at the Temple, and the family fleeing down to Egypt to escape Jesus being murdered, took place some months apart.

Thus, the two accounts of Matthew and Luke are complimentary. There is no contradiction about baby Jesus’ life threatened in Jerusalem, as some have claimed.