How do I delete/remove a home screen web link?
Web clips added by a profile can be set to not be able to be deleted with the X, therefore hiding the X when in edit mode. To remove a web clip added in this way, you need to remove the profile from Settings &rarr; General &rarr; Profiles.

You can also try disabling restrictions if they are on. Go to Settings &rarr; General &rarr; Restrictions and switch it off, then try removing the web clips.