Preserve a query string at the end of a short URL (bit.ly | YOURLS), and preserve it on the resolved URL

Question

• Im trying to create a simple snippet using HTML, JavaScript, and PHP.
• A search bar, and a button.
• Just type something into the search bar, click search and have it search google.com.
• The button adds a query string to the end with the text entered.
• Example: if you type "dogs" into the search bar, and hit search, it will load a new tab, and go to "https://www.google.com/search?q=dogs"*

Here is the problem

• I am using a URL shortner (like bit.ly), but self hosted on my own server called YOURLS.

• My current snippet is now rendered useless because it does not preserve the query string from..

  • search bar string > short link > back to long url.

Recreating the problem

1. I find a link to shorten: https://www.google.com/search/
2. I shorten: https://www.google.com/search/ to https://sho.rt/Google/
3. I then create my snippet with the https://sho.rt/Google/ URL

</form method="GET" action="sho.rt/Google" _lpchecked="1">
  </input type="text" placeholder="What would you like to find on google?" name="?q=">
  </button type="submit" id="submit">Search</button>
</form>

• When button is pressed, it will will add the query string to: https://sho.rt/Google/?q=dogs but once it converts back to the destination address https://www.google.com/search/ ...the query string disappears. It should show: https://www.google.com/search/?q=dogs

Can any of you think of a clever solution to this problem?

• How do I use query string on a short link...and preserve it on the destination address.

• Can this be done in HTML, JavaScript, or PHP? ...Or do I need to adjust something within the self hosted URL shortner "YOURLS"?

You are likely thinking, why even add the addition step of shortening the URL...here is why...

• I have a public HTTPS website (password protected & only accessed by me) with HTTP links on it (hosted within an intranet - secure). This causes a mixed-content warning message since chrome does not understand the HTTP links within an intranet and not public. I found a little workaround...I used a URL shortner tool (it takes an HTTP link, shrinks it, and also adds HTTPS to the short link). I then replaced all the HTTP long urls on my site, with the short HTTPS links, and my site no longer has mixed service, and is secure. This solves 99% of my problems. All that's left is getting my short URLs to preserve the query string to the long URL.