Hurricane Dorian, a category five storm, has sustained winds of up to 180mph (285km/h).
A "life-threatening" storm surge of 23ft (7m) is also predicted in places, officials warn.
The hurricane is moving slowly westwards and may hit areas of the eastern US seaboard.
The US states of Florida, Georgia, North and South Carolina have all declared states of emergency.
Areas of the Abaco islands were reported to be underwater after the storm hit the Bahamas' Elbow Cay soon after midday (16:00 GMT) on Sunday.
The storm also battered Grand Bahama island with high winds and torrential rainfall.
Bahamas residents posted footage showing floodwaters engulfing some homes after high winds had torn their roofs off. Videos also show capsized boats floating in floodwaters filled with debris.
Dorian is expected to continue to move over Grand Bahama Island into Monday.
It is due to move closer to Florida's east coast late on Monday and through to Tuesday night local time.
Media captionHurricane Dorian seen from the International Space Station On Sunday, US National Hurricane Center (NHC) director Ken Graham said the Bahamas would be under major hurricane conditions for 10 hours or more.
Authorities closed some airports in the outlying islands, but the main international airport remained open on Sunday.