How can I quickly delete multiple messages at once in Discord?

Add Mee6 bot to your Discord server. One of its moderator options is a !Clear command.

Usage:
!clear @someone Delete the messages of @someone in the last 100 messages.
!clear xx Delete the xx last messages. Maximum of 1000. Higher numbers may take a while and can crash your discord on mobile devices.

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Alternatively, if adding bots isn't an option, you can make manual deletion a bit easier by holding down Shift. This will skip the confirmation box that usually displays when deleting messages.