That’s very strange. Typically, the widget panel instantly updates the database entry when you remove a widget. From my experience, I can think of four possible explanations.
1). WordPress is broken
Wordpress v 3.9.1 has broken widgets. This probably isn’t the case, since it works on all of my own personal installations. Still, there could be a conflict somewhere that just hasn’t surfaced yet. It could also be that it’s unique to you. Maybe some of siteground’s code has some sort of odd conflict with v. 3.9.1. Maybe there were errors setting up the database. But when you tell me that you’ve reinstalled WordPress multiple times, and the issues are still persisting, I can’t rule it out that there’s a bug in WordPress. Again though, it’s working almost everywhere else, so let’s consider some other things that might be causing the issue.
2). Siteground issues
There may be an issue with Siteground’s installation. I’d keep this in mind if you bought one of their “Easy new WordPress launch”. If you’re not downloading the WordPress installation from www.ads-software.com, and uploading it to the server yourself, there’s a very big chance that misconfigurations are beyond your ability to address. There’s an equally good chance that Siteground’s plugins and themes are simply out of date now. Remember that they have built and maintain their own WordPress themes, and so it is up to them to keep them compliant with current coding standards. A theme that hasn’t been updated for even a single year can have many lines of code that were perfectly acceptable when the theme was last updated, but which have since been rendered obsolete. Such themes can easily break the functionality of WordPress. I’m not sure that’s what’s happening, especially since you tell me that you’ve changed to multiple different themes and had the same result, but just know it’s a possibility.
3). Conditionally triggered default widgets
There is a conditional statement on your theme that displays a set of default widgets when your widget areas are empty. This is actually normal for theme developers. They will check to see if you have widgets assigned to a sidebar. If not, several predetermined widgets just show up without any extra steps from you. You can tell if this is the case because the widgets should be different across each theme. If you change themes (with the widget areas empty) and different widgets appear in the sidebars–you know this is what’s happening. I don’t think this is your issue though, since the widgets keep popping up in your widget areas in the WordPress admin area. If these conditional widgets were the culprit, I would expect to see the widget area in the admin page empty, but still have widgets showing on the front page.
4). Caching (most likely)
What seems most likely to me is that your pages are being cached and redisplayed from previous versions. I noticed that in picture #2, at the very bottom, it appears that you have some sort of built in plugin, or theme feature called SuperCacher. Caching means that instead of computing an entirely new web page every time a user visits your web page, the server saves certain pieces of information that it thinks will not change very often (this is actually perfect for static sites that DON’T change, but less so for dynamic sites like blogs). The problem with caching is that sometimes, the server keeps on sending this old, cached information even when you make a change to the site. Browsers have their own caching as well, where they save images and code from a website you’ve already visited to make your next visit that much faster.
My recommendation is to clear your browser cache, and then play around with the settings in your theme SuperCacher. If that doesn’t work, then some next steps you could take are as follows:
1). Use myPhp admin on your hosting provider’s cPanel to take a look at the database to isolate whether or not the values from the widget table are being saved properly. Clearly, this is overly technical, and can likely be skipped–the following URL leads to instructions about how to access that information (don’t change anything in the database if you go).
https://www.ads-software.com/support/topic/where-is-widget-data-stored-in-the-database?replies=3
2). Manually install WordPress to see if the issue is with Siteground’s one-click WordPress service.
3). Switch hosting providers–bluehost and hostgator are two services that I know work for certain. You could try them.
4). You can try reading through the changelog for WordPress 3.9.1, to see whether or not they updated anything having to do with the widgets. Actually, since you’re not familiar with technical goings on, I’ll take a look and see what changes have been made.
5). If you go through all of these steps, and it still doesn’t work, then I would download a legacy version of WordPress and install it manually on your host provider.