Go to YouTube.com, click on your profile icon at the top right of the screen and press Settings.
Go to the Video Manager section and press History in the sidebar. Then click "Clear all viewing history" and finally click "Pause viewing history" to stop YouTube from recording your future activity.
Next press Search History in the sidebar. Then click "Clear all search history" and finally click "Pause search history to stop YouTube from recording your searches.
Of course at any time you can click the Resume buttons start again keeping search and video views history.