Police have blamed Islamist group Jemaah Ansharut Daulah, which is affiliated with Islamic State.
The church bombing family - whose father was named a local JAD leader - and two other families linked to this week's attacks knew each other and were part of the same Koran study group, police said.
The attack occurred on Wednesday morning at the Riau Police Headquarters in Pekanbaru, the capital of Riau Province, when a minivan, with five terror suspects aboard, attempted to enter the police headquarters' gate.
"I'm outraged and very saddened by these acts carried out by cowards - they have no humanity", said Pebby Magdalena, who joined a demonstration in Jakarta in support of the bill.
Four attackers were shot dead at the scene Wednesday and another suspect who fled was later arrested, police said.
Indonesia's deadliest terror attack was in 2002 on the tourist island of Bali when nightclub bombings killed 202 people, including 88 Australians.
The weekend attacks were the worst in Indonesia - which starts the holy fasting month of Ramadan on Thursday - since the 2005 Bali bombings, which claimed the lives of 20 people.
Indonesia's security forces have arrested hundreds of militants during a sustained crackdown that smashed some networks, and most recent attacks have been low-level and targeted domestic security forces.
"They were part of the Islamic State of Indonesia (NII) network in Riau, which is part of the ISIS network in Indonesia", he said, referring to Negara Islam Indonesia, also known as Darul Islam, a group that aims to establish an Islamic state in Indonesia.
The next day, militants attacked Surabaya city police headquarters.
"They had the same teacher and they regularly met for Koran recital every week", said East Java police chief Machfud Arifin, adding that police were pursuing the teacher for questioning. Police responded by shooting the alleged attackers.