India and Bangladesh began a massive clean-up Thursday after the fiercest cyclone since 1999 killed at least 88 people, leaving a trail of destruction in its wake.

Cyclone Amphan flattened houses, uprooted trees, blew off roofs and toppled electricity pylons, while a storm surge inundated coastal villages and wrecked shrimp farms vital to the local economy.

The United Nations office in Bangladesh estimates 10 million people were affected, and some 500,000 people may have lost their homes.

But the death toll was far lower than the many thousands killed in previous cyclones - a result of improved weather forecasting and better response plans.

The disaster has raised fears, however, that overcrowding in storm shelters will exacerbate the spread of coronavirus.

India's West Bengal reported 72 deaths - including 15 in the capital Kolkata - with state premier Mamata Banerjee saying: "I haven't seen a disaster of this magnitude." "This is the worst cyclone to hit the state since the one in 1737 when thousands lost their lives," she told reporters.

Sixteen people have died in Bangladesh, according to the official death toll.

Improved weather forecasting meant Bangladesh was able to move some 2.4 million people into shelters or out of the storm's direct path, while India evacuated some 650,000.

At least 10 million people were still without power on Thursday afternoon in the worst-hit districts of Bangladesh, said rural electricity board chief Moin Uddin.

The storm levelled more than 55,000 homes - most made of tin, mud and bamboo - across Bangladesh, junior disaster management minister Enamur Rahman told AFP.

Kolkata residents woke to flooded streets, with part of the city's yellow taxi fleet up to their bonnets in water in one neighbourhood and many areas without power.