NEW DELHI — A crushing cyclone barreled up the Bay of Bengal on Tuesday, heading for a swampy stretch along the border of India and Bangladesh and threatening to unleash 165-mile-an-hour winds and massive floods when it makes landfall on Wednesday.

       As the cyclone, Amphan — categorized by Indian meteorologists as equivalent to a Category 5 hurricane — neared the coastal areas, hundreds of thousands of people in India and Bangladesh were bracing for the worst and had started moving toward emergency shelters.

       In the eastern Indian state of Odisha, the authorities have fewer shelters to work with because many have been turned into Covid-19 quarantine centers. Indian officials are now struggling to evacuate people and prepare for floods and destruction while still under a partial lockdown to fight the coronavirus. Humanitarian officials are worried that by packing people into shelters, coronavirus infections could spread even further.

       Satya Narayan Pradhan, the chief of India’s National Disaster Response Force, said the incoming storm could “wreak havoc.”

       “We must take it very seriously,” he said.

       Indian officials said the storm was one of the most dangerous super cyclones to hit India in decades, since a cyclone in 1999 killed more than 9,000 people. That storm packed winds of more than 170 miles per hour, devastating many states along India’s coast.

       Since then, the authorities in India and Bangladesh have significantly improved their emergency response measures, drafting meticulous evacuation plans and building thousands of sturdy emergency shelters, some of which can accommodate several thousand people each.

       Last year, the Indian authorities whisked more than a million people out of the path of a huge storm, deploying millions of text messages, tens of thousands of volunteers, nearly 1,000 emergency workers and television commercials, coastal sirens, buses, police officers and public address systems.

       Many scientists believe that climate change and warmer temperatures are making these super storms even worse.

       In Bangladesh, officials said the storm could bring slashing rains to the muddy, wooden shacks of about a million Rohingya refugees living in Cox’s Bazar. Those refugees fled ethnically driven massacres in Myanmar in 2017 and have been rendered stateless, stuck in limbo in squalid camps in Bangladesh that have been flooded time and again.

       Many people in Bangladesh, apparently, are not heeding the calls to evacuate and move into emergency shelters, despite being told about the risks.

       “There is a sense fear among people,” said Selim Shahrier, a station manager at a community radio station in southwestern Bangladesh. “They hesitate to leave their belongings.”

       Cyclone Amphan is expected to make landfall on Wednesday afternoon, and the storm surge is likely to inundate low-lying areas.

       “Our lives have always been filed with fear,” said Arjun Mohanty, a teacher in the Bhadrak district of Odisha, where the government had turned a shelter into a quarantine center for suspected Covid-19 cases. “First it was coronavirus, now the storm wants to kill us.”

       In eastern India on Tuesday, emergency crews dressed in orange jumpsuits prowled coastal areas, blaring messages from megaphones that urged people to move into shelters as soon as possible. Indian television channels showed footage of the crews moving from place to place, as behind them, the sea was whipped into a white-green froth.

       Indian officials also sent fleets of buses to scoop up vulnerable people and bring them to the shelters, which are stocked with water and food.

       The storm is likely to cause extensive damage to crops, houses, power lines and vegetation, Indian officials said.

       Fierce winds and intense rains have already pounded the southern Indian state of Kerala, uprooting trees, knocking down electricity poles, ripping the tiles off the roofs of many homes and damaging a famous temple.

       Eashwari Thampan, a shopkeeper in the Kottayam district of Kerala, said she was sitting at home with family members when a tree crashed on their roof. Her family, she said, ran for their lives.

       “The wind was so strong it felt as if it was going to take us with it,” Ms. Thampan said. “We thought all of us would die.”