The RAC found only 36% of the 2,000 UK motorists quizzed knew offenders face six penalty points and a £200 fine.
Some 41% believed more visible law enforcement is needed, and 22% advocated stronger penalties.
The punishment was doubled from three penalty points and £100 last year.
This made it enough to ban those with less than two years' experience.
To get back behind the wheel, these offenders have to retake both the theory and practical parts of the driving test.
The new rules apply in England, Scotland and Wales.
Some readers who contacted the BBC said there needed to be more publicity about the tougher laws, while others said they weren't enforced strongly enough.
Martin Tillbrook, from Bracknell, said: "Firstly, the penalties are not widely enough publicised. Why not put roadside signs up to remind people?
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