Nick here from Shopify.
I looked into this with our theme support team. They mentioned that this is more than likely because you are using a third-party app to load the shipping bar. Apps load in after the theme has finished loading so there’s no way to speed this up, unfortunately.
From experience, it looks like you are using the Debut Theme which has it's own shipping/ announcement bar natively in the theme. I took a screenshot from an example of my own test store which you can see below:
To get to the settings to change that you need to take the following steps:
Store Admin > Online Store > Themes > Customize > Sections > Header > Announcement bar
If you are using a different theme than Debut, can you let me know what it is and I can see if they have a native announcement bar also.
All the best, Nick