Big-headed is a term often used to describe someone who is arrogant or conceited. However, there are other synonyms that convey the same meaning. One of them is "egotistical", which refers to someone who has an excessive focus on self-importance. "Narcissistic", on the other hand, is another synonym that refers to someone who is overly self-centered or obsessed with their own appearance or achievements. "Haughty" and "pompous" can also be used to describe someone who is arrogant, as they suggest an attitude of superiority and disdain towards others. Other synonyms for big-headed include "vain", "cocky", "presumptuous", and "smug".