medator Hi. With so many option I would suggest using Falang. At least it's version of language switcher will leave You on the same page and not redirect to frontpage.
What language is the banner set to be shown on? When I open the link I'm in Turkish language, but even switching to English does not show the banner, so I would assume it's not published in right module position. With so many templates I believe they will use different module positions so it's easy to lost track.
At least You have menus working right, that was the hardest part 🙂