I have implemented search in my SharePoint 2013 site. It is returning an error as "Sorry Something went wrong" for anonymous users. I have also done full crawling for the site after enabling the anonymous access.
I used to solve this issue by turning on anonymous search results cache, it's under "setting" in CSWP (you can configure it in CSWP or check the link below for more advanced tricks.)
If it works for logged in users and not for anonymous only then you have to run a powershell script that enables anonymous again for these lists.
I have written a blog post about this: