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
Sleet and snow for some parts of Scotland. Elsewhere in the UK will be cold, cloudy and with some showers.
At least 52 people have been killed after a fight broke out between two rival drugs gangs in a Mexico prison.
Lord Coe has accused Nestle of hypocrisy following its decision to withdraw sponsorship from the IAAF Kids' Athletics Programme.