You can start fresh by clearing the site's local storage in your browser. This erases everything: caught Pokemon, shinies, notes, progress. It's permanent unless you have a backup.
Step 1
Before you do anything, go to the Tracker and Export your data. Save the JSON file somewhere safe in case you change your mind.
Step 2
In Chrome: press F12 to open Developer Tools, go to the Application tab, find Local Storage in the sidebar, click the pokepedia.com entry, and clear it.
Step 3
In Firefox: press F12, go to the Storage tab, expand Local Storage, find pokepedia.com, right-click and Delete All.
Step 4
In Safari: go to Settings, Privacy, Manage Website Data, find pokepedia.com, and remove it.
Step 5
Refresh pokepedia.com. The Tracker will be completely empty. All grey sprites, all stats at zero, all progress bars reset.