The other day I was using Twitter again. But I got so annoyed by all those promoted tweets in between, that I opened the inspector to investigate, whether I could hide them. Note, that this could still lead to ad tracking for you.

I shared the needed CSS snippet, but want to preserve it here as well:

.promoted-tweet {
  display: none;
}

That's it.

Then I looked into the notifications view where even more unwanted tweets were shown. This is enough to hide:

.js-activity-generic {
  display: none;
}

You'll need the stylus web extension to use it.