I am using the promoted links web part. Each item on the list is security trimmed. As normal reader of the site you can not see the links. How can I hide this message to avoid confusion?
You could use the audience targeting feature and only show the webpart to the audiences that are allowed to view the content:
On the "advanced" section of the web part tool pane you can specify a target audience. Make sure to include any groups that have access to the underlying content in your promoted links list. With this feature configured, the webpart will not appear on the page for users that are not part of the "target audiences"
An audience can be identified by using a SharePoint group, a distribution list, a security group, or a global audience. The Web Part appears only to the people who are members of the audiences that you specify here.
https://support.office.com/en-us/article/Target-a-Web-Part-to-a-specific-audience-d4076339-dc25-48be-9a2b-ffafdf33e9d8 (article for SharePoint Server 2007, but feature works the same way in 2013 and O365)