The number of people reporting alleged breaches of Covid rules has been rising in recent weeks, a police chief said.
Suffolk's Assistant Chief Constable Rob Jones said it was accounting for one in 20 calls to the force and officers had been told to "step up" enforcement.
He said: "The time is right. Patience is running out in communities."
Over the weekend, 39 fines were handed out in Suffolk and 31 in Norfolk to those found to have ignored restrictions.
Mr Jones said his officers had been stopping drivers, cyclists and walkers to ask why they were away from their homes.
"The vast majority were out for legitimate reasons but some, particularly on Saturday morning, had gone further out to the coast, perhaps for a walk and stopping at a cafe," he said.
"Those are the kind of cases where we intervene."
England is under a national lockdown, meaning people must stay at home and can go out only for limited reasons such as food shopping, exercise, or work.
Penalty notices can be issued by police with fines starting at £200.
The issue of travelling for exercise was highlighted at the weekend after police in Derbyshire fined two women who had driven five miles from home to take a walk - a penalty that was later dropped.
Norfolk's Chief Constable Simon Bailey gave his interpretation of the rules: "You should remain in your town, village or part of the city where you live.
"Local means leaving your house, ideally on foot, to go for a walk - not driving miles and miles to get some exercise when it's not necessary."
Policing minister Kit Malthouse said: "We are urging the small minority of people who aren't taking this seriously to do so now, and [are illustrating] to them that if they don't they are much more likely to get fined by the police".