As you can see now its next to the fairy symbol, but what happens on the phone design is that the user icon disappears and only shows search and cart, so my suggestion is moving the search bar to the other side of the header.
I appreciate that you have let us know which theme you are using for your online store, that is very helpful! Looking into the Dawn theme, I noticed that changing the search icon on the header section of the online store is not available using Theme Editor. However, it is possible to make a change to the theme’s coding to have the search icon displayed where you would like.
Since the Dawn theme is designed and supported by Shopify, we can use free design time for Shopify themes to allow our internal teams to make the changes on your behalf.
If you are interested in using the design time, you will need to login to your store using the Contact Us page. This will allow us to securely verify you as the store owner before speaking directly to a Support Advisor about the changes you would like to make.