There is probably too little information as of now, but I'll start an answer to build from.

Both font instructions work well in my Chrome browser, so you seem to have a conflicting font installed. Do you see 4 good looking variants of Georgia font in C:\Windows\Fonts\Georgia? They should be installed with Windows 7, but may have been corrupted. See if the font Georgia works in WordPad (search for "whether the problem occurs in a new document" in this guide). There are also instructions how to repair a broken install, try that at your own risk. I recommend doing a backup of the Georgia folder before you remove it.

Update: Added proposed solution and workaround at the bottom of the post.

There is probably too little information as of now, but I'll start an answer to build from.

Both font instructions work well in my Chrome browser, so you seem to have a conflicting font installed. Do you see 4 good looking variants of Georgia font in C:\Windows\Fonts\Georgia? They should be installed with Windows 7, but may have been corrupted. See if the font Georgia works in WordPad (search for "whether the problem occurs in a new document" in this guide). There are also instructions how to repair a broken install, try that at your own risk. I recommend doing a backup of the Georgia folder before you remove it.

Workaround

If installing Georgia font (see investigation section) didn't help, I've prepared a workaround that should be fairly easily applied, even though this is a bit techie to understand fully. (Also be careful to trust internet downloads in general.)

• Install Stylish: https://userstyles.org/help/stylish_chrome
• Install dedicated style: https://userstyles.org/styles/172723/use-arial-on-all-stackexchange-sites
• Refresh.

Both font instructions work well in my Chrome browser, so you seem to have a conflicting font installed. Do you see 4 good looking variants of Georgia font in C:\Windows\Fonts\Georgia? They should be installed with Windows 7, but may have been corrupted. See if the font Georgia works in WordPad (search for "whether the problem occurs in a new document" in this guide). There are also instructions how to repair a broken install, try that at your own risk. I recommend doing a backup of the Georgia folder before you remove it.

Both font instructions work well in my Chrome browser, so you seem to have a conflicting font installed. Do you see 4 good looking variants of Georgia font in C:\Windows\Fonts\Georgia? They should be installed with Windows 7, but may have been corrupted. See if the font Georgia works in WordPad (search for "whether the problem occurs in a new document" in this guide). There are also instructions how to repair a broken install, try that at your own risk. I recommend doing a backup of the Georgia folder before you remove it.

The standard answer to any strange computer problem is to restart computer. It's a great suggestion, because it strangely often works and it requires no technical knowledge. Since history shows that you've had the problem for months, I'll skip that though.

It has been clarified that it is only appearing so for you while others (at least in this case) do see your highlight as intended.

It is still a question if only your text appear bold or if it is the same on other users' posts. This answer has "especially when the author" and some more in bold, if you want to verify if you see that.

When you say that ELU but not ELL is affected, I agree that it's surprising. I haven't looked if they use the same fonts, but I would assume so. I don't know any site specific font override setting, but there are browser plugins that can do it. There are also plugins that sometimes destroy pages accidentally (e.g. ad-blockers and such).

Use another browser for ELU. Try Firefox or Chrome to begin with. If that doesn't work it is likely a problem outside the browser or plugins.

If my suggestions didn't help, or if they did so you can provide more info, the next step is to do deeper investigations on the problem. Ask a techie you have around to help. I'd start with looking at "source view" in the browser debugger of each page to see if there are font differences coming through, and if the markup is there as intended or stripped somehow.

The standard answer to any strange computer problem is to restart computer. It's a great suggestion, because it strangely often works and it requires no technical knowledge. Since history shows that you've had the problem for months, I wont ask you to do that.

It has been clarified that it is only appearing so for you while others (at least in this case) do see your highlight as intended. You're not doing anything wrong in your posts, we have to look beyond that.

Lambie now more clearly indicated that it also affects other users' posts. This answer has "especially when the author" and some more in bold, if you want to verify if you see that.

When you say that ELU but not ELL is affected, I agree that it's surprising. It is however explained by the use of different fonts on the sites. One could say the the two sites are designed to look different, although your problem was not the intended behavior. 

I don't know any simple site specific font override setting you may have accidentally done, but there are browser plugins that can do it. There are also plugins that sometimes destroy pages accidentally (e.g. ad-blockers and such).

Use another browser for ELU. Try Firefox or Chrome to begin with. If that doesn't work it is likely a problem outside the browser or plugins, but sometimes different browsers display pages differently, because of how they are coded.

If my suggestions didn't help, or if they did so you can provide more info, the next step is to do deeper investigations on the problem. Ask a techie you have around to help. I'd start with looking at "source view" in the browser debugger of each page to see if there are font differences coming through, and if the markup is there as intended or stripped somehow.

Chrome debugger (on my Windows 10 install) shows that web coding (CSS on each site) instructs text of type ".post.text" to have "font-family: Georgia,Times New Roman,Times,serif;" on ELU, but "font-family: Arial,"Helvetica Neue",Helvetica,sans-serif;" on ELL. They both override "body.ask-page.unified-theme" that have "font-family: Arial,"Helvetica Neue",Helvetica,sans-serif;". This hidden instruction tells the browser which font to use, on ELU starting with any installed variant of "Georgia", falling back to "Times New Roman" etc. if not available.

Both font instructions work well in my Chrome browser, so you seem to have a conflicting font installed. Do you see 4 good looking variants of Georgia font in C:\Windows\Fonts\Georgia? They should be installed with Windows 7, but may have been corrupted. See if the font Georgia works in WordPad (search for "whether the problem occurs in a new document" in this guide). There are also instructions how to repair a broken install, try that at your own risk. I recommend doing a backup of the Georgia folder before you remove it.