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
The news that Jeremy Clarkson, Richard Hammond and James May are to present a new show on Amazon Prime has been met with a mixed response.
Authorities say Ai Weiwei failed to disclose a previous "criminal conviction" - though his supporters say he was never even charged.
The supermarket made the change after getting feedback from customers who said they felt under pressure as they finished their shop.