Although it's different in each state, most highway bridge rails used to contain either poles or elevated concrete slabs. Very few were just solid concrete. Then, in the mid to late 1970s (some places a little earlier), solid concrete walls resembling Jersey barriers took over in practically every state as the predominant bridge rails. Why? Was there some study that came out around this time that determined these to be safer?
I've done some research on New Jersey barriers a few years ago and came across this article by the FHWA.
The basic operating principal of the barrier is to dissipate some energy and redirect and reorient the crash to be parallel to the barrier by lifting the front tyres to reduce friction between the tyres and asphalt in order to assist the redirection. Many configurations and tests were done and many improvements were made over the years, as also explaned in the article.
This article from Mental Floss is a bit less technical and explains the history a bit more colloquially.
As for bridges specifically, the same safety principles will apply. Many states have their own bridge design manuals which will specify the barrier based on the bridge location and use case and many of the higher safety barriers are designed similar to the jersey barriers with some decorative twists.
I found New Jersey's bridge design manual and section 23 sheds some light on the barrier selection criteria, but many states' manuals are similar.
The oldest standard I could find pertaining to barrier selection based on traffic use case is NCHRP Report 230 from 1981. This was subsequently replaced by NCHRP 350 which seems to be in the process of being replaced by MASH (Manual for Assessing Safety Hardware); if not already.
Older guidelines as early as 1962 are mentioned here but I couldn't find them to see what they specify and how.
I hope this sheds some light on the topic.