So far, I see almost the same thing that @Mittineague sees when I click the row of language flags (I obscured the logo name on the image).
The Google language bar does not appear in Firefox or Chrome (older version) as it does in your screen shot. Can you tell us please describe you development/test environment? browser(s) and version(s)? Perhaps that will help us understand the difference between your screen shot and ours.
The following observations may be helpful as you continue building the site.
I feel sure that over the years that you have been visiting us, someone has mentioned the value of validating your code as development progresses. At this time, the HTML validator, https://validator.w3.org/nu/ , reports 56 errors and 61 warnings. Some of the errors are very significant and could well be the reason that the code does not behave as expected.
You might notice that the horizontal menu intrudes in the Logo at some widths. The font-size in my browser is set to 120%, so that might be consideration; however, @Mittineague's screen shot shows a bit of intrusion, too.
There is no HTML tag at the top of the page, yet there is one at the bottom. The validator barked at the two
<head> tags, too... and there are duplicate IDs in use.
I DO recommend that you start your troubleshooting by validating the code on your site. I can't imagine anyone feeling inspired to try to make anything work as desired when the HTML is improperly structured. Improperly structured HTML cannot be counted on to work consistently across the wide assortment of browsing devices that we have nowadays. That's just the way it is.
I'm sorry I cannot give you a plug-n-play drop-in answer to the Google translator issue.