Purpose of building bridge is to cross the river; so bridge across is used in the context of travelling/moving. E.g., Robison drove on the bridge across the river. If movement is not to be emphasized, then "bridge over the river is correct: e.g., A bridge has been built over the river.

The purpose of building a bridge is to cross the river; so "across" is used in the context of traveling or moving. E.g., Robison drove on the bridge across the river. 

If movement is not to be emphasized, then "bridge over the river" is correct: e.g., A bridge has been built over the river.