We’ve been saying it for years: the huge increase in recorded rapes in this country is because victims are more confident about reporting it. Women have read about Jimmy Savile, they’ve heard about well-known men going to prison and they trust the police more than they used to. But the trend is so relentless that I think we now have to consider a much more alarming possibility – that more rapes are being committed.

First, the figures. The number of rape claims – from both adults and children – recorded annually by the police has more than doubled in the past four years, from just over 16,000 to almost 36,000, according to figures just published by HM Inspectorate of Constabulary. I’ve got used to arguing that these increases are a good thing because we need victims to go to the police, to get justice for themselves and to protect other women.

All of that is true, but it doesn’t look to me as though it’s happening. There is a long-running argument about how to calculate the percentage of rape reports that end in convictions but the actual figure – only 2,689 this year – tells its own story. While that number is slightly up on the 2,414 convictions four years ago, over the same period the proportion of reported cases that led to conviction – already stubbornly low – has dropped by half, from 15% to 7.5%.

Most people have yet to realise quite how extensively the internet is being used to plan and commit sexual offences

I’m not dismissing the idea that some victims have more confidence in the criminal justice system, post-Savile. But I’m worried that the assumption may hide an actual increase in the number of rapes and serious sexual assaults. If the conviction rate had kept pace with the increase in reported cases, there would have been 5,400 convictions this year – more than 2,700 higher than the actual figure.

I’m not alone in being concerned. Women read the news, they watch TV and they know that men who commit rape are unlikely to be convicted and sent to prison. Even if senior police officers and prosecutors have dramatically changed their approach to women who allege rape, there is still a wider culture of disbelief that is evident during high-profile trials.

If you think victims are treated more sympathetically by the media these days, I suggest you read reports of rape trials in the Daily Mail. The paper regularly splashes on stories about men who have been found not guilty, going into prurient detail about the accuser’s “past” – it recently described one young woman as having “an extraordinary sexual appetite” – and failing to acknowledge that false accusations are rare.

Then there are inexcusable delays in the criminal justice system. Long waits to get to court mean that victims have to live with anxiety about giving evidence and fear of intimidation, especially if they knew their alleged attacker. I would never argue that women shouldn’t go to the police but it’s the beginning of a lengthy process, and there isn’t anything like enough support for women who are going through it.

I’m just not convinced that we can go on explaining the entire rise in rape reports by saying that victims are more likely to make a complaint. The possibility of an actual increase is starting to be discussed among senior police officers, prosecutors and organisations that work with victims, and there are a number of factors that might explain it. After all, it isn’t just victims who read angry articles by campaigners about how easy it is to get away with rape.

It is a fact that most men who commit violent sexual attacks on women (and other men, in a small number of cases) will never be brought to justice. When I think about how long sexual predators like Savile got away with their crimes, it is inconceivable to me that there aren’t serial rapists walking round free today, entirely unperturbed by the (distant) prospect of ever being arrested.

If anything, technology has made the whole thing easier. Earlier this year, a man called Jason Lawrance was jailed for life for raping five women and attacking two others he met on the dating site Match.com. Lawrance, who was 50 and married, was a serial rapist who preyed on women who were divorced or widowed.

His is an extreme case, but the internet makes it easy for rapists to invent attractive biographies for themselves and gain the trust of potential victims. Earlier this year, the National Crime Agency revealed that the number of people reporting a rape with someone they met via a dating website had risen sixfold in five years, albeit from a low base.

We also know that the internet has created opportunities for paedophiles to contact underage children, concealing their real age and persuading victims to meet them without parental knowledge. Most people, I suspect, have yet to realise quite how extensively the internet is being used to plan and commit sexual offences, including rape and serious sexual assaults on children.

I hate the idea that more individuals may be undergoing the horror of a sexual attack. The implications are alarming for everyone who wants to live in a country where such violence is beyond the pale. But I don’t think we can go on taking comfort in assumptions, however reassuring, that may turn out to conceal a much grimmer reality.