Why did Jesus leave as soon as he heard that cousin John was imprisoned? Didn't he visit him in jail?

It would not have been possible for Jesus to visit John in prison. Since Jesus was considered associated with John's movement and was followed by disciples who had once been those of John, even preventing himself at the prison would have been making a statement of open rebellion.

Open rebellion was not what Jesus came for. Just like Jesus fled when they crowd tried to seize Jesus and have him made king, just as Jesus said to Pilate, "my kingdom is not of this world," It simply was not Jesus way to present himself as a threat to Herod. His mission was simple - to be the Lamb of God who takes away the sins of the world.

John's mission was very much at his end...

Cousin John was an humble servant, as he himself testified publicly when Jesus came to him to be baptised...

Don't put our ordinary relations as between relatives, with the relations between the Creator and all His creatures...

He is the Master, He knows what He doing... and we not (you humans)...

Angels don't expect like humans, the answers from God... but orders.. learn from us to obey, to be submissive...

not ask questions, but fulfill His commandments...

and be holy... as John... He understood when his desciples brought the answer from Jesus: "Tell to John... the blinds see, the cripples walk, and to the poors it's preached the Gospel"...