Making more progress in mapping Foster City, CA (USA), it's become evident that when creating ways of roads with bridges or tunnels (or any other segment that needs to be tagged in another way), it's easier to make the whole road one contiguous way, tag it with the correct highway type and street name, and then go back later to add in any bridges.
Insert any points needed to make the bridge begin and end at the right place, and then split the way on each side of the bridge, and then add the bridge and layer tags to the newly created way.