How did the Tet Offensive alter the course of the Vietnam War?

The Tet Offensive, in many ways, failed to achieve the ends for which it was designed by the North Vietnamese. Indeed, the North Vietnamese sustained heavy casualties, and the attacks on South Vietnam did not ultimately stir up the level of anti-American feeling the North Vietnamese had hoped for. Originally, they had conceived of the offensive as a means of breaking up the alliance between the Americans and the South Vietnamese by demoralizing the South Vietnamese and forcing the Americans to leave. This did not happen.
Despite these apparent failures, however, the Tet Offensive did deliver a strategic blow on the part of the North Vietnamese in a way that would change the course of the Vietnam War. The Vietnam War had never been a popular war in the US. However, news had been spread at home that the war would soon be over and that the North Vietnamese could be readily defeated if only the US hung on for long enough. Television footage showing the massive scale of the fighting between the US and the North Vietnamese, however, eroded public support for the war still further: the imagery was often extremely graphic. What successes the Vietcong did have, furthermore, were particularly demoralizing for the American people, who watched as the Vietcong attacked the American Embassy in Saigon.
Generally, the North Vietnamese, who lost ten times as many soldiers as the US and South Vietnamese combined, suffered worst in the fighting, as they simply were not militarily strong enough to achieve the success they had aimed for. Their failure to inflict huge South Vietnamese losses, furthermore, meant that the South Vietnamese did not move to dissolve their alliance with the US. However, the great success of the Tet Offensive lay in the fact that it made clear to those at home in the US that the Vietnam War was not "almost over." Indeed, after the offensive, US Commander William Westmoreland put in a request for over 200,000 new men in order to launch a counterattack. This was a significant number, which could be interpreted at home as evidence of fear. It also suggested that Westmoreland was by no means certain the war could be easily won.
The Tet Offensive by no means meant the end of war in Vietnam. However, it encouraged Lyndon B Johnson to begin de-escalating American participation in the region, minimizing the extent of bombing in response to public outrage and growing dissent among American politicians. Johnson announced that there would be no more bombing about the 20th parallel. The war at this point moved from being simply divisive, to one that was more generally unpopular than popular. The Tet Offensive brought American public attention to what was really going on in Vietnam, and the reality turned them away from it decisively.
https://history.state.gov/milestones/1961-1968/tet

https://www.history.com/topics/vietnam-war/tet-offensive