I spent lot of time to fix a strange issue while working with SharePoint 2013 project. Issue was a user having Full Control permission in SharePoint Top level site was not able to create a sub-site. Each time user tried to create either a "Project Site" or "Team Site" user got “Sorry, you don’t have access to this page” or “Access Denied” error. This was the identical behavior for all the users having Full Control permission.
1. It appeared SharePoint was ignoring the Full Control access permission.
2. User with site collection administrator permissions could create sub-sites.
3. New site collections on the same web application operated normally without this issue.
1. Internally, SharePoint manages a hidden list named "TaxonomyHiddenList". URL of the list is - "[sitecollectionURL]/lists/taxonomyhiddenlist".
2. Generally, All authenticated users (everyone) has Read access to this list.
3. As a site collection administrator user, when I checked permissions of the list, I found none of user or group were having permissions to this list as you can see in below screenshot.
Resolution: The solution was simple once identified. We should assign read access to all authenticated users (Everyone) operating within the site collection.
I had spent more time to fix the issue as it was a difficult one to identify, Hope this helps you out to overcome it more efficiently.