I have created a custom webpart and deployed it in SP2013 site.when I am trying to add the web part it throws an error like, "The type could not be found or it is not registered as a safe". - (PFA).
Your Web part is not correctly declared as Safe in the web.config. Do you see the web.config SafeControl entry in the wb.config? Was is added automatically?
The class SharePointProject1.WebPartName.WebPartName is not declared public i guess.
The DLL is not in the GAC neither in the app_bin folder. How do you deploy the solution?
I've experienced this issue several times when deploying custom web parts. If retracting and redeploying doesn't resolve the error (which it likely will not), you can try doing an iis reset.