Following up on the King’s Peace series: Why would a king not require that conflicts—including trials—between great houses be settled by the king or his designee? That would prevent houses from legally striking at each other with arrests and farce trials. If Catelyn were required to remand Tyrion to Robert, some of the conflict might have been avoided, or would have happened with less justification. Such a system would also help the king to show favoritism and win allies.

A king who made it their mission to expand judicial kingship would indeed have done that, and if you look back through the history of all of the more successful monarchies, you’ll see that a large part of their expansion of power came from expanding judicial authority, eliminating the nobility’s judicial privileges, and preventing private wars and vendettas through the extension of the concept of the royal peace. 

But unfortunately the Targaryens largely neglected this authority in favor of its monopoly on dragons.