how do i stop chrome showing old searches in the address bar

Shift plus Delete on a highlighted suggestion removes just that one, handy for the embarrassing ones. For all of them: Settings, Privacy and security, Clear browsing data, tick browsing history, time range all time. To stop it saving new ones: Settings, You and Google, Sync and Google services, turn off autocomplete searches and URLs. If you're signed into Google the searches may also be coming from your Google account's Web and App Activity rather than Chrome itself, that gets cleared from the account's My Activity page.