If you haven't mess with anything deeper and assuming it is a casual joomla installation with your ACL additions, then it is definitely something you have not done right or ignoring in setting up your usergroup and access level.
It can be any combination of if your usergroup is or is not a child of other usergroup, and that while registered usergroup has viewing access to your Custom Access Level, your custom Usergroup has not. Add in the mix that the menu items of your menu might have also ACL settings and depending on the layout of the Module, the module will not be displayed at all, if there is no access to its content.
Make a step back from the website and read a bit the documentation about implementing ACL in Joomla, in order to get a better understanding.
I would suggest to start your experimentation with the Joomla ACL slowly, doing small changes and testing what is working and why, as it can be really confusing and easily a website can become messy.