The menu on my site goes down when the browser size is reduced and at a resolution of less than 1024x768, the site is here: shororus.ru. Site is on WordPress.
Have a look at getting a mobile menu plugin such as WP Mobile Menu (https://wordpress.org/plugins/mobile-menu/) so that you can set at what pixels the mobile menu should display. By customising the amount of pixels in the required plugin, as you shrink it down it should just convert to a mobile menu and not force the menu off of the header.
Hope this helps.