Rereading Psalms, I'm struck by the psalmist's obsessive pleas to punish sinners and destroy enemies, compared to Jesus' love for sinners and admonition to "turn the other cheek." Why is this?

Your observation is apt. Anyone reading the Psalms may be troubled to come upon passages where the writer implores God to punish his enemies in specific and sometimes violent ways. If we are honest though, in our heart of hearts, we may at times resonate with these passages. These are called the “Imprecatory Psalms,” calling for curses on the enemy (Psalms 5, 10, 17, 35, 58, 59, 69, and others).

It is important to note that there is something good about the psalmist’s recognition of evil and his desire for justice and righteousness. It is right to hate evil and to want it avenged. In an essay in his book Reflections on the Psalms (chapter 3), C. S. Lewis doesn’t excuse the Psalmist’s practice, but he does argue that the sense of moral indignation behind these curses is better than the indifference of those who live looking the other way, those who fail to call out what is wrong. However, while the psalmist’s curses are understandable human responses, they are not examples of what is right. We must examine our feelings, catching bitterness and hatred, asking God for the ability to forgive, and then trusting Him for the outcome.