Find the URL that's actually ranking
Search your full name in quotes, then your name plus the case caption, and copy the CourtListener link out of the result exactly as it appears. Send the URL that ranks. If the link ends in a document number, climb back up to the docket it belongs to and send that instead.
Check who else is holding your case before you spend a minute on CourtListener. Look at the issuing court's own website, then search Justia, Casetext, Leagle, vLex, anylaw and Google Scholar by name, and if CourtListener isn't on your first page of results, this isn't your problem to solve today.
Send the de-index request
The request goes through CourtListener's contact form, which opens with Case Removal Request already selected in the topic menu. It asks for a name, an email, a subject and a message, then a captcha. You're sending this one by hand.
Paste the URLs. Don't describe the case. Here's a message that does the job: "I am the person named in the pages below. Under your content removal policy, please block them from search engine indexing." Then one URL per line and nothing else.
Budget half an hour. Nearly all of it goes to finding the right URLs rather than writing the message. Anyone invoicing you a CourtListener removal fee is charging you to send a contact-form message. Don't hunt around their site for a removal page either, since the old one is gone and the contact form is the whole route.
What happens after you send it
Blocking the opinion page also blocks the docket it sits under, so one URL does both, but it doesn't run the other way. If a docket and an opinion page both rank, list both. Nobody can give you a date. CourtListener publishes no turnaround, no acknowledgment window and no service level, and their policy warns up front that they have no control over any search engine, so your case may sit in the results for many months, if at all. Don't pay anyone who promises you a date.
Send it once and leave it alone. There's no status page to check and no second address to try. Spend the waiting time on the copies sitting on the other sites.
If it's still on your name later
Search your name again once a month. Say the CourtListener link is gone and your case is still on the first page: you're looking at another site's copy, because a block never reaches some copies, and every one of them takes its own request.
If the page comes back, send the CourtListener request a second time. They make no guarantee that a blocked record stays blocked, and every block granted without a court order is theirs to reverse. A repeat costs the same few minutes.
If you need it actually deleted
It takes two things, both required: the information deleted from the public record, and a court order demanding that Free Law Project do the same. Their content removal policy states both conditions. It's the only route to deletion they recognize.
An order aimed at a court clerk or a state agency doesn't reach them. They've refused a federal judge's removal order before and fought it rather than complying, so if you're paying a lawyer for an order, budget for the chance it gets contested instead of honored.
With an expungement or redaction order, expect anonymization rather than deletion. Names get replaced with initials or black boxes, and a permanent note at the top of the document says the change was made. One order also covers one document, so name every filing you want touched, because they won't change the others unless the order specifically requires it.
None of this touches the court's own record. Taking down an aggregator copy seals nothing, expunges nothing, and changes not one line in the clerk's file or in PACER. That takes a motion in the court that issued the filing, which is separate work.
What not to do
Don't sue them. Free Law Project says it has been sued over publishing court documents and has never lost. One plaintiff's push to bury her case ended with the entire docket unsealed and the fight written up in public, and their terms also fix venue in California, so a plaintiff anywhere else is litigating away from home.
Don't file a credit-report style dispute either. Free Law Project isn't a consumer reporting agency, and their terms of service bar using the site for credit, insurance, employment or housing decisions. There's no dispute to file here, and nothing ages off on its own.