The Battle of Ap Bac showed the extreme inefficiency and unreadiness of the south Vietnamese army. That was one of the earliest events.

Other than that, it was more gradual until the Tet Offensive. Increasing news coverage showed the brutality of the war and how it was fought. It also gave Americans a better idea who they were fighting instead of simple labels like "communists."

The Tet Offensive in 1968 was decisive in turning the public against the war. Despite the North Vietnamese under General Giap losing a massive number of troops and being unable to complete their objectives, it showed the public that the enemy was unwilling to quit and would fight on regardless of the consequences. TV footage of a Viet Cong soldier being shot execution-style in the street and of American soldiers and CIA operatives attempting to regain control of their own embassy through force were seminal examples of how shitty the war actually was. Keep in mind this offensive came after a government announcement that the war was going well and that the enemy would be soon beaten.

Nixon's continuation of the war despite his claims to wanting to end it did nothing to help either. Or his being a crazy crook in general.