Currently, there are on average over 30 shark attacks a year in the United States, with about one death every two years. These numbers were almost certainly lower over 100 years ago, but there's no way that there were no shark attacks in the years before 1916. Why was a shark attack thought of as an unlikely possibility?
Expanded from an earlier answer of mine
Part I
Jaws, both the novel and the book, certainly did shift cultural attitudes towards sharks as dangerous predators. However, neither Spielberg nor Benchley invented this image of the predator out of whole cloth. Neither did the Matewan attacks invent the concept of a shark attack. There was a long association within modern Western culture that associated sharks with with predation on humans that long predated the New Jersey attacks.
The harpoonist Queequeg in Melville's doorstopper of a novel Moby Dick commented that "de God wat made shark must be one damn injun," in a chapter in which the sharks go into a feeding frenzy over a whale carcass. Queequeg's opinion of the shark mirrored much of contemporaneous Western thought on the shark in the nineteenth century. The shark mixed both mystery and unbounded savagery. Verne's 20,000 Leagues Under the Sea tried to demystify some of the shark's behavior and biology, but a shark's attack on Captain Nemo's underwater oyster farm still played to tropes of sudden attack by an animal hell-bent on consuming human flesh. Nor were these depictions of sharks limited to print. Winslow Homer's 1899 The Gulf Stream depicts the horror of a shark attack at sea. The Matawan shark attacks of 1916 certainly popularized the fear of sharks. John Singleton Copley's painting Watson and the Shark depicted the shark attack on Brook Watson, one of the Lord Mayors of London, when he was a boy.
Copley was heavily influenced in his portrait by one of the more famous maritime paintings of his era: Théodore Géricault's painting Raft of the Medusa. One of the most famous shipwrecks in the tail end of the Age of Sail was the French frigate Medusa which became beached in 1816 off Mauritania. The ship's crew constructed a raft for the 146 passengers and crew who could not fit in the ship's launches, and the intention was for the launches to two the raft to shore. The lines broke and the raft became adrift with little provisions. Thirteen days later, the raft was rescued and only fifteen of the raft's passengers were alive. The cause of death ranged from suicide, fights, attacks by sharks, and, in what proved most shocking for European contemporaries, cannibalism. Much like the relationship of the Indianapolis to Jaws, the travails of Medusa's crew became immortalized in by Géricault's painting. While sharks may not be present in the painting, they were one of many hazards present in writings on the shipwreck. One of the more notable aspects of these tales was that the sharks arrived after the crew indulged in cannibalism.
One of the survivors, Henri Savigny wrote of the sharks surrounding the raft and preventing the survivors from collecting fish, which were the only available form of sustenance, and their jaws even managed to straighten a bayonet hook. Savigny noted as fights began to break out:
it was a number of sharks which came and surrounded our raft. They approached so near, that we were able to strike them with our sabre, but we could not subdue one of them, notwithstanding the goodness of the weapon we possessed...The blows which [a crewman] struck these monsters, made them replunge into the sea; but a few seconds after, they reappeared upon the surface and did not seem at all alarmed at our presence. Their backs rose about 30 centimeters above the water: several of them appeared to us to be at least 10 meters in length.
Savigny's timing of the appearance of the sharks, just when human relations were breaking down, connected his account into the larger symbolism of the shark in European Romantic literature which emphasized the animal's innate baseness and predatory savagery. Becoming like a shark for a human meant preying on his fellow man. William Cowper's 1785 poem The Task invoked the shark in its meditations on the growing sprawl of London:
London ingulfs them all! The shark is there,/ And the shark's prey; the spendthrift, and the leech/ That sucks him; there the sycophant, and he/ Who beheaded and obsequious bows/ Begs a warm office, doom to a cold jail/ And groat per Diem, if his patron frown.
Not surprisingly, abolitionists latched onto this metaphor of men becoming as sharks in both prose and poetry attacking the slave trade. James Thomson's poem Summer had this passage:
Increasing still the terrors of these storms, /His jaws horrific arm’d with threefold fate, /Here dwells the direful Shark. Lured by the scent /Of steaming crowds, of rank disease, and death, /Behold, he, rushing, cuts the briny flood, /Swift as the Gale can bear the ship along; /And, from the partners of that cruel trade, Which spoils unhappy Guinea of her sons, /Demands his share of prey—demands themselves. /The stormy Fates descend: one death involves / Tyrants and slaves; when straight, their mangled limbs /Crashing at once, he dyes the purple seas / With gore, and riots in the vengeful meal.
In Thomson's account the slave-traders were no different from that of the shark, they were indeed "partners" in human misery. The abolitionist Thomas Clarkson contended that the slave trade altered sharks' migratory patterns as they followed the slavers:
These voracious fish were supposed to have followed the vessels from the coast of Africa, in which ten thousand slaves were imported in that one season, being allured by the stench, and daily fed by the dead carcasses thrown overboard by the voyage.
In true Swiftean fashion, the Scottish radical James Tytler's pamphlet The PETITION of the SHARKS of AFRICA asked the House of Lords to consider the sharks when discussing the slave trade. The sharks were aghast that there was even talk of abolishing an institution which so easily afforded them their favorite food, human flesh. Should the Lords give into "this new-fangled humanity", then the population of sharks would regretfully diminish.
Although obviously embellished for dramatic effect, the abolitionists' use of the image of sharks feasting on castoff slaves was attested to in various accounts of both slavers and in the colonies. Samuel Robisnon recalled in his memoir that as a teenager serving on his uncle's slave ship that the sharks clustered underneath the ship and the crew were afraid to stick their hands in the water lest they get bit. The entry for the White Shark in Thomas Pennant's 1776 encyclopedia British Zoology derived a number of its observations on this particular shark's from slavers' accounts, including one gruesome tale in which a slaver captain showed his cargo that they could not escape by jumping over by lowering a corpse into the water and pulling out the shark-eaten remains. Reports of the Royal African Company occasionally mentioned that sharks swarmed embarkation areas for slaves. William Smith, a surveyor to the Royal African Company, recalled that sharks swarmed the boats off the African coast in hopes of finding prey and would later note that as slave ships dumped their corpses, the "sharks give such due Attendance, that the Corpse can no sooner touch the Water, then it is immediately torn to Pieces, and devoured before our Faces." Jamaican newspapers reported in 1785 that the arrival of slavers also came with overgrown sharks, presumably fed from slave corpses. While notions of the slave trade altering the global migratory patterns of sharks are unlikely, the floating reef of a slave ship did likely provide sharks a human-based meal.