"나는 싸웠다" is considered an awkward sentence in Korean because the speaker does not say who he or she fought with. Koreans would normally say something like 나는 그와 싸웠다, which means "I fought with him." In English,"I had a fight" may sound all right, but Korean normally requires that you mention with whom you had the fight.
It takes two to tango, it takes two to fight, and it takes two to do many other things, as well. There are certain Korean verbs that require a partner or an opponent to perform the action, and among these verbs, some require that if you mention one partner or opponent, you mention the other. Often you mention that partner or opponent by using one of the "and" conjunctions, that is, 와, 과, or 하고. Here is a list of some of those "two-to-tango" verbs: