Only a fifth (21%) felt that free schools should not have to follow the national curriculum, the survey for Channel 4 News found.
The programme revealed the poll results as it emerged that another free school, in Bradford, is in trouble.
While 81% of people agreed that it should be compulsory for free schools to employ qualified teachers, three quarters (75%) agreed that qualifications do not necessarily guarantee a good standard of teaching.
More top news
Lucy Beale's killer will be revealed live on air during a special week of EastEnders episodes to mark the soap's 30th anniversary.
Spandau Ballet reunited on stage for the first time in five years as the band performed at the premiere of a film documenting their career.
From tomorrow, tax discs will no longer be needed to show drivers have paid their vehicle tax. Richard Pallot reports on the end of an era.