Why did many americans turn against the war they had once supported between 1964 and 1969?

The Tet Offensive of 1968 was a turning point, and the influence of the media was felt in that turning point as well.

The number of American households that owned a television set rose dramatically in the 1960s.  In the 1950s, less than 10% of homes had a TV, but by 1966 that had risen to 93% of households.  Most Americans now were getting their news from television, and Walter Cronkite was the most trusted anchorman on TV news.  When the North Vietnamese and the Viet Cong launched the Tet Offensive in 1968, Cronkite, known as "the most trusted man in America," offered a television editorial that shaped the nation's mood.  Cronkite said in that broadcast on February 27, 1968:  "It seems now more certain than ever, that the bloody experience of Vietnam is to end in a stalemate. To say that we are closer to victory today is to believe, in the face of the evidence, the optimists who have been wrong in the past.  To say that we are mired in stalemate seems the only realistic, if unsatisfactory conclusion. ...It is increasingly clear to this reporter that the only rational way out then will be to negotiate, not as victors, but as an honorable people who lived up to their pledge to defend democracy, and did the best they could."

In addition to Cronkite's reporting, there were also images coming from the war front that had shown a "credibility gap" between what the US government had been saying about the war and what was actually happening there.