Recently I used content by search web part in SharePoint 2013. Web part was working perfectly fine for all the site collection administrators. However when any other user was browsing the page, they were getting "Sorry, something went wrong".
After analysing all the queries and permission, I found that display templates I had uploaded did not have any published version. I published the major version and it started working.
In case you are getting same error, please ensure if display templates are published.