Plus, your arrest record is a matter of public information, I'd be shocked if they did considering in America (if you are from America) the Freedom of Information Act gives these sites rights to post your criminal history. It's like sites that post the residence location of Sexual Predators (though I have no problem with that and SHOULD be a matter of public record, FoIA or not).
Isn’t publishing my mug shot a violation of my rights?
No, it is not. Originally collected and distributed by Law Enforcement agencies, Booking records are considered and legally recognized as public records, in the public domain. Furthermore, while it may be you or someone you know in the mugshot, the ownership of these records is public
Taken from the FAQ's page of mugshots.com
They do have an "Unpublishing Policy" but read the FAQ's if your case falls into certain category then yes you could get them removed.