Short answer
You can download any past conversation from the History screen using the three-dot menu next to it or by selecting multiple chats and clicking the Download button.
Full explanation
AskTuring lets you save past conversations for offline reference or record-keeping. To download:
Go to the History screen from the left-hand menu.
Find the conversation you want to download.
Choose one of these methods:
Download a single conversation: Click the three-dot menu next to the conversation and select Download.
Download multiple conversations: Check the box next to each conversation, then click Download at the bottom of the screen.
The file will be saved to your device in the selected format.
Tips
Keep downloads secure if they contain sensitive information.
Renaming the file after downloading can make it easier to organize and locate later.