append to url and refresh page

this should work (not tested!)

var url = window.location.href;    
if (url.indexOf('?') > -1){
   url += '&param=1'
}else{
   url += '?param=1'
}
window.location.href = url;

Shorter than the accepted answer, doing the same, but keeping it simple:

window.location.search += '&param=42';

We don't have to alter the entire url, just the query string, known as the search attribute of location.

When you are assigning a value to the search attribute, the question mark is automatically inserted by the browser and the page is reloaded.

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