I think people overestimate the strength of the US and underestimate the strength of the Entente in North America, even Turtledove himself.
Here's how I see the big-picture overview in 1914:
The East: Let's assume the Confederates don't play offensively. The US needs to advance 100 miles on a narrow front stretching from the Blue Ridge to the Bay, all the while making several major opposed river crossings, and doing so in a combination of urban terrain (Alexandria and Fredericksburg being the main ones), and densely wooded areas (the Wilderness/Chancellorsville). They also need to do this against an enemy which likely has planned for such an attack for 50 years, and is thus well dug-in. Even if US forces outnumber the Confederates 2:1, they're looking at a very bloody and slow-going campaign that would take a year or two to reach Richmond at best. And, for reference, it took Grant about a year to advance from Spotsylvania to Richmond in 1864-1865, and that was against an enemy he outnumbered by around 3:1 with comparatively minimal pre-planned defenses. You're asking the North to add an extra 50 miles of ground to that, on top of an enemy with better numerical parity and well-prepared defenses. It won't be pretty. Historically we know such terrain was a quagmire for WW1 generals, as evidenced by battles such as the Meuse-Argonne Offensive, Belleau Wood, etc. so I don't see the US getting very far within the first few years of a war.
The West: Luckily for the North, Louisville sits right at the border, and there is plentiful infrastructure at it to undertake a large operation. The main obstacle, however, is that they would have to make an opposed river-crossing against a built-up urban area in ground they tried attacking once before, so the Confederates have likely prepared plenty of defenses in anticipation of a future assault. It will be bloody and slow going, but I think the North would be able to pull it off by the end of 1914. The problem then becomes, however, how they advance from there. Contrary to popular belief, the terrain is still narrow, being shorter than the length of the Western Front from the Mississippi River to the Appalachian Mountains. It also involves multiple large river crossings, mountainous terrain with very little in the way of infrastructure, and an enemy that enjoys decent interior lines. Still, the North's numerical advantage would likely tell, and I imagine they'd be able to at least capture Nashville by 1915. The question really is what the hell they do when they get to Chattanooga. They'd not only have to make an opposed river crossing, but they'd also have to do so against a major urban center flanked on either side by flat-topped Lookout Mountain (which provides excellent artillery positions) and Missionary Ridge. Flanking around the city opens them up to enfilading fire from Lookout Mountain, which runs all the way to near Fort Payne, Alabama. Flanking to the east would mean they'd have to cross a series of mountain ridges to get back to the railroad supply line that runs south from Chattanooga, through Dalton, and on to Atlanta. Even if the North manages to capture the city, they'd likely be stuck in the valley getting pummeled by the Southerners occupying the heights around it day in and day out. The South IMO could hold that position almost indefinitely, assuming they're able to bring up reinforcements and ammunition from Atlanta without much trouble. What you'd be looking at thus is something akin to Isonzo; the North bashing its head against the fortress of Chattanooga for a year or more while racking up casualties and not much else.
I could easily see a scenario where, by the autumn of 1916, the Northern public simply becomes tired of taking hundreds of thousands of casualties in the Wilderness of Virginia and the mountains of Tennessee and elects a pro-peace candidate who signs an armistice with the Entente.
