This probably-non-peer-reviewed document by a fisheries scientist gives a lower estimate circa 2000 but I can't parse out if his ~55M estimate includes targeted fishing or "only" by-catch:
Considering an average weight of 15 kg per individual, the total number of skates, rays and sharks killed by fisheries would be close to 90 million fish per year, but this is probably overestimated as 15 kg for the average weight might be a bit low. However, if we consider only the sharks, then the figure is closer to 55 million or probably less (again, due to a conservative estimate of average weight per fish of 15 kg; many commercially important and common bycatch shark species easily attain average weights between 50 and 100 kg per fish). Sharks are only about ½ of the reported catch (i.e. 380,000 t), they represent most of the high-seas bycatch (lets say 390,000 t/y), but only a minor part of the likely 200,000 t/y or so of skates, rays and sharks taken and discarded (i.e. unreported) in coastal bottom trawl fisheries (i.e. 50,000 t/y). Thus, the total of 55 million sharks by dividing the total kill of 820,000 t of sharks by 15 kg per fish. These very rough figures try to consider all sources of shark catches, the reported ones, the unreported bycatches in high seas fisheries, and the unreported bycatches in coastal trawl fisheries.