The short answer is you probably cannot force the removal of your mugshot. The book in photo taken at the jail is part of the record of your arrest generated by the State and is generally information routinely disclosed to the public. Recently, websites and publications have begun collecting and marketing this public information in new ways. It may seem wrong but there is no legal way to stop the dissemination of this type of information by third parties many of whom will either be or at least claim to be operating under freedom of the press protections.
Mugshots are public record and can be obtained by anyone through a simple freedom of Information act request. I helped someone get their picture off of Mugshots.com, but they had to pay them a fee. It seems like extortion to me, I was essentially told that they were a business and if you paid a fee of X number of dollars they would remove your photo from their database.
I have noticed in my area of Georgia that the commercial publishing of booking or "mug shot' photos is growing and people are interested in the public humiliation of their fellow citizens. The truth is that the booking photo is public record and is routinely released by law enforcement agencies. Because the photos are routinely acquired by the agencies during standard booking procedures, I believe that you will have difficulty in forcing or requiring the website to remove the photo. I know this isn't the answer you were hoping to receive, but I believe it is accurate. Good Luck!
The information provided in this response to a question is not legal advise and is provided only for general information purposes. My response should not be taken as legal advise as no attorney / client representation exists. Additionally, the information given in this answer is specific to the State of Georgia only and should not be applied to any other state.