Depending on the jurisdiction, you may have to make this initial request to the judge or to the public defender's office. The judge and/or public defender's office will look at your reasons for wanting a new court-appointed lawyer, and will also consider whether you've already changed lawyers on this case. A first request is usually more likely to be granted than any subsequent requests.
You can also ask your lawyer to request that they be relieved from the case. If the judge grants their motion, a new attorney may be appointed to represent you. But before agreeing to the attorney's request to withdraw from the case, get confirmation that a new attorney will be appointed to you. In some instances, the judge may interpret your agreement with the request to mean that you are able to hire and pay for a private lawyer, and no longer have a need for a court-appointed lawyer or public defender.
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