Lai, who is to be inaugurated on Monday, won January's election, but his Democratic Progressive Party (DPP) lost its majority in parliament.
The main opposition party, the Kuomintang (KMT), has more seats than the DPP but not enough to form a majority on its own, so it has been working with small Taiwan People's Party (TPP) to promote their mutual ideas.
The opposition wants to give parliament greater scrutiny powers over the government, including a controversial proposal to criminalise officials who are deemed to make false statements in parliament.
The DPP says the KMT and TPP are improperly trying to force through the proposals without the customary consultation process in what the DPP calls "an unconstitutional abuse of power".
"Why are we opposed? We want to be able to have discussions, not for there to be only one voice in the country," DPP lawmaker Wang Mei-hui, representing the southern city of Chiayi, told Reuters.
Lawmakers from all three parties were involved in the altercations, and traded accusations about who was to blame.
The KMT's Jessica Chen, from the Taiwan-administered Kinmen islands that sit next to the Chinese coast, said the reforms were to enable better legislative oversight of the executive branch.
"The DPP does not want this to be passed as they have always been used to monopolising power," she told Reuters, wearing a military-style helmet.
Taiwan is a rambunctious democracy and fighting does on occasion take place in parliament. In 2020, KMT lawmakers threw pig guts onto the chamber's floor in a dispute over easing US pork imports.
The clashes raise the prospect of more turmoil - and parliamentary conflict - ahead for Lai's new government after it takes office.
"I am worried," said the DPP's Wang.