In fact the Confederacy severely overestimated support for them in Union states.
The plan that the Confederates had was to invade North through Kentucky and Pennsylvania and gather volunteer troops from there to mount an assault on Washington DC and end the war fairly early in 61-62. They did not get that support, as much as they were expecting it.
The North had a more reasonable plan of choking the Confederacy out with an Embargo of their ports (The Anaconda Plan), taking control of the Mississippi to essentially force the Confederacy to face three different fronts (Virginia, the Mississippi river, and then Western Texas).
The War went on as long as it did by an large because the Union had incompetent leadership until the Summer of 63 with George Meade and Ulysses S. Grant gaining prominent roles in the armies of the Potomac and Tennessee respectively. The Confederacy utilized their better generals well in the early going but frustratingly did not have the support they hoped for, and with the embargo on their ports, could not get supplies internationally despite some vague dialog with the British Empire and France.
It came down to the Confederacy having a smaller army but better leadership early on, who time and time again failed to utilize strategic advantages they had and threw their men into a meat grinder against the North rather than the Washingtonian strategy of "retreat, retreat, retreat, hit them when in great position, then retreat again".
The North had the larger force, better equipment, and industrial power to smash the Confederacy, but their leadership sucked. Had Meade taken charge in 1861 rather than McClellan, I think things may have turned out much differently
