Problem: By default, there is no feature in salesforce we can delete all debug logs on single click of a button. As you can see in below screenshot, you have ‘Delete All’ button but when click on this button, it deletes only a bunch of logs.

Solution: You need to run below command in javascript console of your browser from debug log screen using below steps-
- Go to Setup->Debug log.
- Now press Ctrl+Shift+I to open browser inspector.
- Now click on console and paste following code in this window; and now wait and watch till all debug logs get deleted.
setInterval(
function() {
if (document.querySelector('span[id*="deleteAllStatus.start"]').style.display === 'none') {document.querySelector('input[value="Delete All"]').click()}
}
,250);
Demo: