The TMC, headed by fierce Modi critic Mamata Banerjee, has accused the BJP of attempting to import divisive sectarian politics into the state, which has a large Muslim minority.
On Tuesday shots were fired outside several polling stations during clashes in areas outside the state capital Kolkata, leaving at least 20 people injured, police said.
Dilip Ghosh, a senior local BJP official, told AFP that the mother of one party supporter was killed by TMC supporters. The TMC denied the allegation and police said three people were arrested.
“The surge in attacks by TMC supporters has eclipsed everything this time,” Ghosh said.
Voters were being intimidated into staying away from polling stations “in hundreds of villages”, he added.
The votes were being held even though India is experiencing a sharp rise in coronavirus infections, with several regions imposing restrictions on activities.
Because of the need for extra security, West Bengal has spread its election over eight phases culminating on 29 April.
The north-eastern state of Assam, home to 32 million people, was also holding the last out of three phases on Tuesday with the BJP vying to hold on to power.
The state is polarised along ethnic and religious lines, with immigration from neighbouring Bangladesh one of the biggest campaign issues.
A “citizenship list” in Assam state in 2019 left off almost two million people unable to prove they were Indian, many of them Muslims, a process many fear the BJP wants to roll out nationwide.
The southern regions of Kerala, Tamil Nadu and Puducherry also went to the polls, with local parties dominating the race but the BJP hoping to make inroads.
In Tamil Nadu an alliance led by MK Stalin -- named by his father after the late Soviet ruler -- was aiming to win power with his DMK party.
In Kerala the Left Democratic Front, a coalition including two communist parties, was fighting to retain power against an alliance led by Congress, the BJP’s main national rivals.
Puducherry, a small former French colony previously known as Pondicherry, has been rocked recently by political defections and resignations leading to the collapse of its Congress-led government.
Results from all the elections, which come two years after Modi was elected to a second straight term, are due on 2 May.