The Weekly Vista has a story about how the name Lost Bridge came about and here's a summary:
The folks living in the Garfield area wanted a more direct route to Eureka Springs, a bridge was proposed by the county to be built at Fishtrap Ford in Fishtrap Hollow. A contract was awarded for a Tennessee contracting company to build an arched cement bridge, 318 feet long, 16 feet wide and 40 feet above the river. The Tenn. company won with a bid of $24,125.
Work was started in June of 1929, material were brought to Garfield by train and hauled on a narrow wagon road to the bridge site.
The bridge was completed in October 1929.
When the bridge was completed there was not funding available to build a road to it, so it became the "Lost Bridge".
The bridge opened in 1934 once funding had been found to build the approaches to the bridge.
On May 7, 1943 heavy rain caused the White river to flood, a log jam and flood waters caused the bridge to collapse.
A low water crossing was built in the aftermath of the bridge collapse and was in use up until the time the site was flooded by the rising water of Beaver Lake in 1964.
Pic of the completed bridge, in it you can see the approaches have not been completed:
Low water bridge, to the left is the washed out bridge: