BAGHDAD — A year after the last American troops rumbled out of Iraq, the two countries are still trying to get comfortable with a looser, more nuanced relationship as the young democracy struggles to cope with political upheaval and the legacy of war.

In the meantime, Iraq is busily pursuing its own interests — sometimes against America’s wishes — as it seeks to balance its position in a precarious part of the world and reestablish itself as a regional power.