At some polling stations, the number of voters had barely reached double figures six hours after polls opened

Turnout appeared to be sluggish as voting got under way in the first ever police and crime commissioner elections across England and Wales in the 41 police force areas outside London.

At some polling stations in urban areas, the number of voters had barely reached double figures by lunchtime on Thursday – six hours after they had opened – but in other parts of the country, particularly in rural shires, voting was brisker, even exceeding some predictions.

Labour sources say the turnout among postal voters appears to have been around 38-40%, which would point to an overall turnout percentage in these elections in the low 20s.

The opposition party has set itself a benchmark of winning "a majority of policing" outside London. This would mean Labour controlling the forces that have a majority of officers or spend the most of the police budget. In practice this means winning the eight to 10 largest urban forces, including the West Midlands and Greater Manchester.

Labour sources say they are "looking very good" in seven of the largest forces but dismiss predictions by some commentators that they are on course to win as many as 19.

In particular, Humberside, where Lord Prescott is standing, is said to be very close. The supplementary vote system means that the substantial support for Conservative and Ukip candidates could result in Prescott facing a strong challenge from second preference votes.

It is also thought that unexpectedly strong funding of two or three independent candidates who have been able to afford a privately funded mailshot to all voters means they could overcome the name recognition problem and break the apparent two-party stranglehold on the election.

A presiding officer at a Leeds polling station tweeted: "First voter came in after 2 hours 30 minutes. Her candidate might be winning." But in other parts of the country there was better news: "Polling station staff said turnout is low so far, but better than some predictions," said a Kent tweeter.

David Cameron tweeted from @David_Cameron: "We're helping police forces be more accountable. I've voted in the first ever #PCC elections. I hope you do too."

A Home Office spokesman said: "Today is a historic day – people across the country have the opportunity to vote for the first ever police and crime commissioners.

"Their arrival will be the most significant democratic reform of policing in our lifetime. PCCs will give the public a real say in how their communities are policed and will use their mandate to cut crime.

"Crime is down, but we recognise that people are still worried about crime and that victims need to be looked after. PCCs will help to reconnect the police force with the public and will focus on what is really important for law abiding people."

However, a new study showing that fear of crime has fallen to its lowest level for eight years may provide a key to understanding why turnout in these elections is lower than might have been assumed.

The national policy monitor based at Essex University says levels of satisfaction with the way the police deal with crime have also been rising in recent years. Eight years ago, 55% said they were unhappy with the way the police dealt with crime and only 45% said they were happy. But those figures have now reversed, with 55% saying they are now happy with the work done by the police to cut crime. It may be that people are not flocking to the polls because they don't think policing is broken.

Professor Paul Whiteley, the monitor's director, said: "It looks as if the reduction in crime that has occurred over the years is actually beginning to pay off."