Hi rogimon
The problem came from layout settings only, on the homepage of our demo site, this template doesn't show the mainbody block on homepage but it's set to display on your site so it pushed the search module to underneath it.
I just updated via: Admin panel > Extensions > Templates > ja_directory - Default (2) > Layout tab > and add the 'Home' menu item into the 'Skip component content' field: https://prnt.sc/tsy3py
It should be ok now, kindly check.