There's no rule defining the minimum heights of bridges.
Many rail bridges, for example, were built over minor roads which, at the time, were only intended to be tall enough to allow a man leading a horse to get underneath. Some of the tracks, which went under those bridges, have now become busy roads. For example, there's a road near me, with a constant flow of traffic in both directions, with an 8'3" bridge. Elsewhere in the country, I've seen bridges over busy roads with heights of around 7'.
Chris