Rape victims in Britain are badly served, Boris Johnson has admitted, as he promised to “stop at nothing to make sure we get more rapists behind bars” and to overhaul the policing of sexual violence after Sarah Everard’s murder.

The prime minister said the rape and murder of Everard by a serving police officer had “triggered feelings of huge numbers of people” about the handling of sexual crimes, domestic violence and rape.

Only 2% of reported rapes in England and Wales result in prosecution, and rape convictions are at an all-time low, while many instances of sexual harassment are not fully investigated – failings the Guardian has highlighted repeatedly. Courts are also experiencing long backlogs in hearing cases after struggling with cuts to the criminal justice system and 18 months of Covid-related delays.

Speaking before the Conservative party conference in Manchester, Johnson said policing and prosecution of such crimes was “going wrong”. Rape victims “need to get a better service” from the police, he said, as he highlighted possible changes in the way evidence is handled and mobile phone data is collected.

Johnson told BBC One’s The Andrew Marr Show: “We will stop at nothing to make sure that we get more rapists behind bars and we have more successful prosecutions for rape and for sexual violence. Because that, I think, is going wrong.”

The government and the Metropolitan police have been under huge pressure over their response to Everard’s murder by Wayne Couzens but have resisted calls for the Met commissioner, Cressida Dick, to resign.

The issue is already overshadowing the Tory conference, which Johnson had hoped would focus on his agenda for “levelling up” the country to reduce inequalities between different regions.

Asked how women could trust the police, when they were being told to flag down a bus if they suspect an officer is trying to harm them, Johnson insisted it was “very very important that people should have confidence in the police”.

However, he acknowledged serious problems in the policing of sexual violence. The prime minister blamed a failure of police and prosecutors to work together well enough to ensure there was a chance of a “decent case being presented”.

He said people and women in particular “know instinctively that something is going wrong”. “We do need to look systemically at not just the Wayne Couzens case but the whole handling of rape, domestic violence, sexual violence and female complaints about harassment all together,” he said.

Johnson rejected calls for a public inquiry into the police’s failings on sexual violence and in the Everard case, saying investigations by the Met and the Independent Office for Police Conduct should be allowed to happen first.

On Friday, the prime minister described the police failure to take sexual violence against women seriously as “infuriating”, pointing to long delays in the criminal justice system that can result in people waiting years for their case to be heard.

The home secretary, Priti Patel, also told the Telegraph that the police needed to “raise the bar” in responding to all violent crimes against women, including those that are sometimes seen as low-level.