It can be done by doing some code customization. please send me a personal message and we can discuss what you'd like
Hi @ZaDu ,
as KetanKumar hinted it can be done. You will find the code in the file header.liquid in the Sections folder of your theme. There you basically can just change the order of the elements.
The problem here is that your theme, Debut, is here quite simple - meaning it does not differentiate much what kind of screen size you have, only the hamburger menu (three lines icon) reacts. So if you reorder your header this will affect all user not only user with small screens (mobile phones). If this is ok for you than it is easy otherwise it is a little bit more work.