When you worry excessively about something, you agonize about it. The mother of a teenage boy might agonize over his safety when he first learns to drive.
Agonize can mean to suffer true pain, but it's generally a mental type of anguish. A student might agonize over what to wear the first day of school, and a boss might agonize about how to break the bad news about layoffs to her employees. The earliest meaning of agonize was "to torture," though now it means something closer to "to torture one's self." The Greek root is a grand one: agonizesthai, "to contend in the struggle."