It is unfortunate to share that SharePoint 2013 is not supported by <IE8 browser versions. In case you got this error: “Object doesn’t support property or method ‘querySelector’ ” while loading the site page that means you have to go to new version of browser.
This error encounters because of init.js file under layouts scripts which uses the java script query selector as shown below:
So getting rid of this error is impossible as Microsoft never checked IE7 compatibility while releasing this new version of SharePoint 2013. In case your customers are bound to use this browser then you have to coordinate with Microsoft guy to handle this issue.
I am not sure but there is one another way to get rid of this. Some people have used some tools like sizzle for same.