Open your browser settings
Go to Privacy or History
Select Clear browsing data
Choose the time range (All time if available)
Check the option for Search history (and optionally Browsing history, Cookies, Cached images and files)
Click Clear data / Delete / Clear browsing data
If available, choose Manage data or Search history and delete individual entries
Sign out of your account (optional) and clear data again if you use synced search history
For Google Search, open Google My Activity and delete items or clear all activity
For YouTube search, open YouTube History settings and clear search history
For Bing, open Bing search history settings and clear search history
