The only way to ""fire"" your appointed public defender is to hire a lawyer to step in and take over the case. Your hired lawyer will contact your appointed public defender.
You cannot simply fire your public defender because you are not the one who put him on the case; the judge appointed him.
In order to "fire" your public defender you have to tell the judge you want a Marsden motion. That would allow you to tell the judge, in private, (private means the public is not allowed and the prosecutor is not allowed in the courtroom), why you think your public defender is not effectively representing you. See People v. Marsden (1970) 2 Cal.3d 118. If the Court relieves the public defender after a Marsden motion is granted, then the Court will appoint a different public defender to represent you. You can hire counsel yourself, but if you wait too long the judge may not let you substitute new counsel.
The answer from the Texas attorney is not entirely correct. In California, a defendant can "fire" a public defender by making a Marsden motion. You tell the judge, promptly, that you are making a Marsden motion. The courtroom will be cleared. The Judge, the bailiff, the court reporter, you, your public defender, and the court clerk stay in the courtroom. The prosecutor and the public are excluded. You state your reasons why you think the public defender is not representing you effectively, the public defender responds and the Judge decides whether to relieve them or not. If so, then you get a new public defender appointed. You can also hire an attorney but the Court has discretion to allow a new attorney to come into the case depending on how long the case has been going on, how close to trial, and other factors. Good Luck.
The answer from the out of state attorney is incorrect. That may be a correct answer in Texas where she practices, but it's not entirely correct here.
The Marsden motion mentioned is the procedure in California to try and get the public defender off your case. If you're successful, the court will relieve the public defender and appoint a new public defender. Marsden motions are sometimes granted because there truly is inadequate representation going on or a substantial breakdown in communication between attorney and client. I don't know if your situation rises to that level just yet, but if you're serious about pursuing this, let your public defender know you want a Marsden motion. They will tell the court and the proceedings will be underway. If your public defender does not inform the judge at the next court date, speak up. You don't want to wait until your trial to complain - it may be seen as too late or simply a delay tactic.
Or you can hire the attorney of your choosing and they can take over, relieving the public defender.