The Conway Arms as shown in the photo was demolished a long time before the fly over was built. In the 1950s there was a new Conway Arms on that same site, known as the flat iron {due to it being a single story building and with been on the V corner of the roads had the shape of a flat iron.}
Ships that pass in the night, seldom seen and soon forgoten
This was the same building as was demolished in the late 1960s when the flyovers were built, and it was indeed often called the flat iron. The picture below is from the Conway Street side, and you can see how far out the single storey part on Claughton Road side protrudes, giving the appearance of a single storey building from that side.