Oh, you meant swinging like swaying. I thought you meant swinging bridge like a swing bridge that swings open and closed. We had both kinds.
What you're describing are the two old bridges over the Cooper River. They were both replaced by the Ravenel Bridge.
http://ravenelbridge.net/papers/Cooper_River_Bridge_US.pdf
TCS and I used to run the annual Cooper River Bridge 10k Run on the old bridge. During the run, the bridge was closed to traffic. With just runners on it it used to bounce. They had to prohibit the Marines from running in time as a group because they made it bounce too dangerously. It was interesting, to say the least.
There was a weight limit that prohibited large trucks on the older of the two bridges (Grace). Also, because it was so narrow vehicles were constantly getting their side mirrors knocked off.
After the new bridge was built they took down the two old bridges.