It’s not your browser.
The bottom line here is that advertising powers the internet. The way those sites pay for their operations is by hosting advertising of various forms.
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Why sites use advertising
Many (if not most) of the sites that you visit are completely free to you. You don’t have to pay a dime to access them. You just have to see that there are ads.
The double underline words that pop up advertisements are one form of website advertising. If you see it only on certain sites, then those sites have this elected to use kind of ad serving technology. It was on Ask Leo! for a while, but I removed it a while back.
It’s nothing more than an option that’s available to website owners to generate some revenue to cover the costs of running their site.
What if it’s malware?
If you’re seeing this on all of the sites that you visit, then it could be malware. For example if you’re seeing it on Ask Leo!, then it almost certainly is, since I’m not using that ad serving technology any more.
But unless this is appearing everywhere, this is typically an option that website owner use to generate some revenue.
If it’s too annoying, then the only solution that I’m aware of is to stop visiting those sites.