Hi Charlie Scheid,
Welcome to our forum!
As I know: Select the X and the contact will be removed from the list and will also be hidden for later use. If you send an email to the contact by replying to an old mail or by typing the full email address on a new mail. After approximately 24 hours, their information will be restored to the list, that is the contact will be unhidden from the suggestion list. It is by design.
There is no method to disable suggested contacts feature and delete its entries permanently on Outlook for Web, this feature is designed to make more convenient using experience, though it doesn't suit for you, your understanding is highly appreciated, if you fairly don't want some contact to appear automatically when typing certain keyword, it is recommended you try to remove it from your contact list, as it is not realistic to remove the contact in your organization's global address list.
Hope it helps.
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