It is possible to win a counter-insurgency campaign...
Yes, sure. Witness the British in Malaya, or Kenya to take just two examples.
However, the Vietnamese situation was complicated by external state actors, the North, the Soviet Union. Even if you could have isolated the southern insurgents from their supporting base, (Either the decent way, like Malaya, or the deplorable way, like in Kenya), they could still have received substantial support from the North, like they did during the majority of the war. So any strategy would have had to be two pronged, both a "hearts and minds" to isolate insurgents from their support in the south (the US didn't have the stomach for a Kenyan solution, which is a good thing), and a military intervention against supporting external regimes and their supply lines. Without the latter part, the insurgents would still have had much too much wind in their sails, for too long.
As it was, they couldn't get popular support, in no small part due to the corruption of the southern regime, and a decisive military intervention was politically impossible. A no-win situation if there ever was one.
And even with the right strategy, counter insurgencies take time. Something the US doesn't have, as their political will, and hence staying power has been, and probably will always be, low.