If the PHP version of your domain is set to run PHP FastCGI, the PHP functions flush(), ob_flush(), and ob_implicit_flush() will not function as expected.
By default, output is buffered at a higher level than PHP (specifically, by the Apache module mod_deflate which is similar in form/function to mod_gzip).
Solution
If you need unbuffered output, you must either use CGI (instead of FastCGI) when selecting your site's PHP version, or contact support to request that mod_deflate be disabled on your site.