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If you are using managed navigation instead of structural make sure that the Navigation Node Type is a Simple Link or Header, the Term driven menu items are not visible to users with only read permissions. You can find more info about it here: http://sharepoint.handsontek.net/2016/05/12/metadata-navigation-is-not-visible-to-users-with-read-permissions/
If you do have Minimal Download Strategy feature enabled than please deactivate it & try it again.
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