You have the right to an attorney which you are charged with a crime. If you can't afford one the State must appoint one. In Washington that attorney is required to meet minimum standards set by the bar and the Washington Supreme Court. You do not have a right to choose which attorney is appointed to represent you. If you don't have faith in your public defender you need to start interviewing private attorneys and find one you can both afford and trust.
You don't have a conflict of interest with the Public Defender on the facts you've stated (because they lost a case representing you). Public Defenders ARE experienced lawyers. If you don't like a particular Public Defender, you can request a different one through the lawyer's office, but you don't have a choice once one is assigned. If you really have a conflict with a particular attorney, like if they don't communicate with you, or fail to do something according to professional rules of conduct, you may have a hearing with the court to determine if there is a legal conflict.
You can always hire your own attorney and many people I know prefer it.
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It depends on the Court and the Judge. If you have a conflict with the public defender's office, and are indigent, most courts have a list of attorneys in private practice who have agreed to take cases on an appointed basis. You can ask the Court to appoint one of those lawyers to represent you. Some Judges will allow you to choose which lawyer you want off the list. But most Judges will simply assign your case to the next lawyer on the list. Maybe you get the lawyer you want, but most likely you don't. The only way to ensure you ge the lawyer you want is to hire one. Good Luck.