I don't know if deleting it every time you exit the browser is a good idea, but perhaps an auto delete after X time. So after it's three days old, or one week old, or two weeks old, or four weeks old, or some sort of custom amount between one day and four weeks old, or something like that.
That way you're not keeping excess history around forever, but yet you have something to go back to if you need it within a certain small timeframe.
Another example is that ProtonMail automatically deletes emails in the spam and trash folders after 30 days. So you don't have to worry about them. When you delete an email, you just delete it and you know that 30 days later it's going to be gone.
