Will making a grass bridge wider increase the speed at which grass covers it?

Yes, making a bridge more than one block wide will increase the speed of the grass spreading. Grass can spread to any adjacent blocks (including diagonals), from one level above to three levels below the "source" block. If you were to build a one-wide bridge, the grass would have to spread from the first block in the bridge to the second, then to the third, etc. However, with a > 1-block-wide bridge, there are multiple blocks that this algorithm could be run on. Also, grass could spread to the block in front or the blocks diagonally from it, allowing for the grass to propagate faster.

Wider is absolutely faster, but diminishing returns suggests only a width of two or three. Also you can simply grow a protruding block of grass and then push it with a piston. This way, you can get the grass where you want it, let grass spread from it, and maybe push it around to 'seed' different parts of the area you want to make green.