Open Instagram and go to your profile
Tap the menu icon in the top right
Select Your activity
Tap Recent searches
Tap Clear all
Confirm by tapping Clear all again
To remove a single search
Open the search tab
Tap the search bar
Find the item under recent searches
Tap the X next to it
To prevent future search history from being saved
Log out after using Instagram on a shared device
Clear app cache on your device if needed