A little more of an explanation is needed. What do you mean by average? By average do you mean typical highway bridges or the average of larger, longer span bridges?
The depth of a bridge is directly proportional to the load and span. Inherent in the load previously mentioned is the bridge width. A 100' bridge carrying vehicular traffic will be shallower than the same span which carries rail traffic. So, you must let us know what the nature of traffic on the span.
Also, there are many ways to span a specific distance. For instance, an 80' span could be accomplished with either wide flange beam or prestressed I-girders. If the span were a little bit longer the bridge may be plate girder or if not too much longer a deeper Prestressed girder. Selection of the superstructure is not always a function of span, but of many factors including maintenance and aestetics.
In general, typical highway bridges could be broken down this way:
Here in the States, a short (30 feet) prestressed deck girder bridge could be as shallow as 30 inches, while longer span bridges (100-120 feet)could be as shallow as 7 feet. Values do not include parapet. Longer span (200) could be 10 feet plate gider or maybe a deck truss. Note: All of the above values assume slab and girder or deck beam.
Many other types of superstructure exist: post-tension concrete box girders, truss bridges ( a 600 foot Warren truss might be 90 feet tall), cable stay (with box grider), suspension bridges(could be double deck) and tied arch bridges.
If you have some specific information, it would help immensely to pinpoint an answer to your question. [sig][/sig]