Mind if I throw in some numbers?
Distance from London to New York: 5,600km
If you built a perfectly straight bridge this long, the end of it would be at an altitude of ~2,100km above the surface of the earth, and an angle of 48.7° to the gravity vector (apparent uphill slope of 41.3°), making it a very tough climb!
How far along this bridge would you need to travel to officially be in space? 1,133km, at which point it would feel like an uphill slope of about 10°.
Would it appear to slope upwards? Well, actually, it would just a bit. Thanks to the fact that air density falls as altitude increases, light bends down slightly, which would make a completely flat surface appear to slope up gradually. How much would depend on the density gradient at the time.
Of course, this is all assuming a round earth. I've no idea what the distance is supposed to be on a flat earth, as there is a lot of conjecture as to the proper layout of the continents, but a flat bridge would follow the surface in this case. Note: it would still appear to curve upward slightly (as would the ground beneath it), thanks to the density gradient of the atmosphere.