According to figures supplied by 23 forces across England, Wales and Scotland, more than 2,000 warrants were issued for specific offences of violence such as murder, causing death by dangerous driving and assault.
A further 423 warrants were granted for sexual crimes including rape. Only partial information was given by 29 forces while two constabularies did not respond to requests.
But extrapolated nationwide, the figures suggest more than 30,000 criminals are being sought throughout Britain.
The figures, supplied to the BBC, are for warrants issued between December last year and last month and cover cases where a defendant failed to appear for trial or for a court appearance.
Bail is usually granted to suspects by police, magistrates or a judge unless they are deemed likely to flee, commit further offences or interfere with witnesses.
On Thursday night, Javed Khan, chief executive of the national charity Victim Support, urged police to "rectify this situation" because they were failing innocent people.
"It is disturbing that there are such a high number of outstanding warrants, particularly for people who are charged with violent and sexual crimes," he said.
"Not only does this create considerable anger and frustration for victims but it means that justice is not achieved and puts more people at risk of becoming victims."
Heidi Olseen, who was raped at a party hosted by a close friend in London in 2008, remains in fear after her alleged attacker went on the run.
Kimamo Wanjunki Kimamo, 25, was charged with raping her and bailed but failed to attend Hendon Magistrates' Court in London in January 2009. He has not been seen since and police have since appealed for information.
Miss Olseen, who has waved her right to anonymity to help other people in a similar situation, told the broadcaster that there "is no sense of justice for me as no justice has been done".
She said: "Something needs to change - especially if you want people to come forward and report crimes, especially serious crimes.
"It seems that [if] the police are so confident in the evidence that they've got, why don't they have the powers to hold somebody until the case comes to court?
"What is the point of the police or the courts? What is their role if in this case nothing has been done about it?"
She added: "Nobody has been punished for it, so where is the justice, because I can't see it? I'm the only one that's been punished - and my friends and family."
Assistant Chief Constable Alison Roome-Gifford, from the Association of Chief Police Officers, defended police forces.
She said many warrants were for minor crimes such as traffic offences.
"What is most important is that police forces have robust systems in place to identify, track and locate those individuals who present the greatest threat of harm to the public," she said.
"When such an individual fails to appear, police forces will prioritise searching for such people and bringing them before the courts at the earliest opportunity."
A Home Office spokeswoman said an outstanding arrest warrant was an operational matter for the police.