Open Instagram
Tap your profile picture
Tap the menu icon
Go to Settings and privacy
Tap Messages and story replies
Tap Show in Chats
Open the chat you want to clear
Tap the person or group name at the top
Tap Privacy and safety
Tap Clear chat history if available
Repeat for each chat you want to clear
To remove a conversation from your inbox, swipe left on the chat
Tap Delete on iPhone or Delete/Remove on Android
Repeat for each conversation to clear your inbox