To delete Instagram messages, you can use the 'Unsend' feature.
To delete an entire conversation on Instagram (mobile), open the app, tap the Messenger icon, long press on the conversation, and select Delete.
To delete an entire conversation on Instagram (desktop), visit Instagram.com in a web browser, log in, click on the Messenger icon, select the conversation, tap the Info icon, and click on Delete Chat.
To delete individual messages on Instagram (mobile), open the app, tap the Messenger icon, open the chat, long press on the message, and select Unsend.