At least three people were reportedly killed by falling trees in three southwestern Bangladesh districts.
Also one death each has been reported from Indian states of West Bengal and Odisha due to heavy rain triggered by the "very severe" cyclonic storm, Xinhua news agency reported.
The storm made landfall near Sagar Islands in Indian state of West Bengal with its status downgraded to "severe" from "very severe" on Saturday night as the gale wind speed dropped to about 115-125km per hour.
It has started crossing coastal Bangladesh districts in a reportedly weakened form. In Bangladesh, it damaged houses, uprooted trees, snapped power and communication lines, disrupted rail, road and air traffic. Hundreds of thousands of people were moved into storm shelters.
"Bulbul" lied over Bangladesh's southwestern parts at around 5:00 a.m. Local time on Sunday and was moving slowly northeastwards.
Bangladesh meteorological department on Sunday morning reported "strong winds and a dangerous storm surge" along the coast after the cyclone came ashore and warned that an inclement weather would persist on Sunday.
They said Bulbul already weakened and has gradually been reducing to a depression.