Background

The Bootstrap theme has an Admin Navbar as well as the Navigation Menu with your site's content on it.

The Admin Navbar contains the search box as well as Login/Logout buttons, and the Admin/Edit buttons if you're logged in as an administrator.

The Admin Navbar can be configured to show at the Top, Bottom (as you have it), or None.

Note that if you select "None" it'll still appear if you log in, so that you can access the admin buttons. None is a useful option if you don't want your users to see a Login button (which could confuse them).

Probable Solution

A newish option in the Bootstrap theme is that you can select Search in Navigation Menu which puts a Search button into the Navigation Menu, instead of showing the search bar in the Admin Navbar.

When you click the Search button the search box appears below the Navigation.

You can currently se an example of this on the Abingdon Athletics or Superhighways site (if you're reading this in the future it may not be the case!)

For your site I suggest trying the Search in Navigation Menu option, and possibly also setting Admin Navbar to "None" if you don't want your users to see a Login button.

Thanks!

Joe