Millions of Americans are missing out on a chance to avoid debilitating fractures from weakened bones, researchers say, because they are terrified of exceedingly rare side effects from drugs that can help them.

Reports of the drugs’ causing jawbones to rot and thighbones to snap in two have shaken many osteoporosis patients so much that they say they would rather take their chances with the disease. Use of the most commonly prescribed osteoporosis drugs fell by 50 percent from 2008 to 2012, according to a recent paper, and doctors say the trend is continuing.

Last month, three professional groups — the American Society for Bone and Mineral Research, the National Osteoporosis Foundation and the National Bone Health Alliance — put out an urgent call for doctors to be more aggressive in treating patients at high risk, and for patients to be more aware of the need for treatment. It followed a flurry of recent articles in medical journals documenting and bemoaning patients’ abandonment of traditional osteoporosis drugs. But osteoporosis experts are afraid their efforts will do little to change minds.

“Ninety percent of patients, when you talk to them about starting one of these drugs, won’t go on,” said Dr. Paul D. Miller, medical director of the Colorado Center for Bone Research, a medical practice in Lakewood, Colorado. “Ninety percent who are on the drugs want to come off. The fear factor is huge.”

Half of those who start taking the drugs stop within a year. Even patients who just broke a hip, which makes another hip fracture extremely likely, are refusing them. In 2011, only 20 percent of patients discharged from a hospital with a broken hip had a prescription for one of the drugs, compared with 50 percent in 2002.

There is little question that fractures caused by fragile bones are a real problem, particularly for women. A 50-year-old woman has a 50 percent chance of having an osteoporotic fracture in her remaining years. The drugs, meant to be started when bone density falls very low and the chance of a fracture soars, can reduce that risk by half, studies show.

But to many, it matters little that the drugs’ frightening side effects are extremely rare. Estimates are that 10 to 40 in 100,000 osteoporosis patients taking the drugs — including alendronate, ibandronate, risedronate and zoledronate — have suffered broken thighbones. Fewer than 1 in 100,000 have had the jawbone problem.

“You only need to treat 50 people to prevent a fracture, but you need to treat 40,000 to see an atypical fracture,” said Dr. Clifford J. Rosen, a professor of medicine at Tufts University who has no association with the makers of the drugs.

Lawsuits over the rare side effects resulted in large jury awards and drew widespread attention. And after reports of these problems began to surface, the Food and Drug Administration requested that the drugs’ labels include a warning about the association.

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