Generally it builds the road on the existing turn, not the following turn.  Does the tile you're on have a road already? Just in case it's not clear, the process is (for a unit w/RoadBuilding ability)

1) Move to a tile without a road

2) Click "Build Road" (this does not consume a turn)

3) Move to the next tile where you'd like a road, etc.

I've found that sometimes it won't build a road if you have 0 movement points left (even though it doesn't consume any movement points) - but it's not consistent.  Usually you can build even after using all your moves.

What would be really nice would be a mechanism to direct a unit to build a road from point A to point B...but I'm not aware of any such functionality in the game.