Johnson had the support of the public before 1968. The US arrived in Vietnam in 1965. That means US forces had three years of fighting with some 500,000 American troops with extensive aerial and naval support.
But those three years had produced nothing but casualties and a stalemate. The war was already on a losing path before the public turned against it.
The public turned against the war in 1968 because Tet showed that after three years, the US and South Vietnamese forces had not even been able to secure the major cities, including the Southern capital, Saigon.