What is the best method to quickly delete old backups on Mac?

You don't sat what kind of backups‽

iCloud?

Manage your iCloud storage - Apple Support

Or Time Machine Backups?

https://www.minitool.com/data-recovery/how-to-delete-time-machine-backups.html

Or Local Snapshots‽

Purging local backups

Please note that although this doesn't affect your remote backup from Time Machine, this will get rid of the redundancy (at least until the next Time Machine backup) that a local backup disk will provide. If you need such redundancy or are worried about the recovery of your data then you would be best served to let macOS determine when to purge these files.

Start Terminal from spotlight.

At the terminal type tmutil listlocalsnapshotdates. 

Hit enter.

Here, you'll now see a list of all of the locally stored Time Machine backup snapshots stored on your disk.

Next you can remove the snapshots based on their date. I prefer to delete them one at at time. Once my "System" disk usage is at an acceptable level, I stop deleting but you can delete all of them if you want to reclaim all of the disk space.

Back at the terminal, type tmutil deletelocalsnapshots YYYY-MM-DD-HHMMSS , where will be one of the dates from your backup. This will be in the form of xxx-yy-zz-abcdef. Try to start with the oldest snapshot.

Hit enter.

Repeat for as many snapshot dates as required

http://www.thagomizer.com/blog/2018/03/27/cleaning-up-time-machine-local-snapshots.html

tmutil deletelocalsnapshots / # deletes all the snapshots