The IFG says legislation will be needed to establish new policies on areas such as customs and immigration.
The extra measures will place "a huge burden" on Parliament and government departments, the think tank says.
But the government said it was confident it could deliver the changes required within the time allowed.
In its report, Legislating Brexit, the IFG says that with the average Queen's Speech announcing only 20 new bills, the introduction of 15 Brexit bills before the UK even exits the EU "will leave very little space for non-Brexit related legislation".
The report comes as Theresa May travels to Swansea with Brexit Secretary David Davis, where she will talk about the "precious union" of the UK.
The prime minister will meet First Minister of Wales Carwyn Jones, as well as local businesses, as she tries to show she is including all areas of Britain in negotiations with the EU.
Mrs May will say: "I want every part of the United Kingdom to be able to make the most of the opportunities ahead."
The IFG report anticipates the new bills will be in addition to the Great Repeal Bill, which will scrap the 1972 European Communities Act that paved the way for the UK to enter the then-EEC, ending the legal authority of EU law.
The IFG - an independent charity that aims to increase government effectiveness - says departments will need "ruthlessly to prioritise" other legislation and find non-legislative routes to get the laws through, particularly given the government's narrow Commons majority.
It warns that this will mean ministers having to achieve a fine balance between giving too little parliamentary scrutiny and too prolonged, in-depth examination of Brexit-related legislation.
The IFG also argued that "a lack of clarity" about the role the devolved legislatures will play in legislating for Brexit could pose a problem.
"The attitude that the Scottish National Party (SNP) takes to the passage of Brexit-related legislation in Westminster could affect the smoothness with which that legislation passes through Parliament if they join forces with the Labour Party and Conservative rebels," the report says.
Dr Hannah White, IFG's director of research, said the government had finite resources to draft new legislation and Brexit bills would take a "big chunk" out of its capacity to legislate in other areas.
"In the first session after the 2015 election the government passed 23 bills," she told BBC Radio 4's Today. "So that's roughly the capacity that there is in government to draft these bills and in Parliament, in terms of parliamentary time, to pass them."
The reality, she suggested, was that the government's domestic priorities would "take a hit in the next couple of sessions".
"The legislation required for Brexit will leave little parliamentary time for anything else - and making a success of it will require a large volume of bills and secondary legislation to be passed by Parliament against a hard deadline."
A government spokesman said it had made clear there would be additional legislation, in areas such as customs and immigration, on top of the Great Repeal Bill but insisted this could be completed within the parliamentary timetable.
"This government will harness the skills and resources across all departments to ensure that the statute book functions effectively on the day we leave - as part of delivering a smooth and orderly Brexit," he said.
The Queen gave Royal Assent to the European Union (Notification of Withdrawal) Bill last week, clearing the way for Theresa May to start talks to leave the European Union.
The Bill allows the prime minister to notify Brussels that the UK is leaving the EU, with a two-year process of exit negotiations to follow. Mrs May says she will trigger the process by the end of the month.
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