I just flag them and move on. Your garbage site doesn't want me reading it, I won't.
The opinion pieces are another story though (pun intended), I generally try to avoid those.
You're confusing accuracy with bias. It is possible to give an accurate account of a story despite having a bias, so long as you're still stating the relevant facts. The bias comes from the interpretation and whether they state the facts in a positive or negative light.
As for biases, WSJ is known to be center-right, NYT is center-left, The Economist is biased towards classical liberalism, etc. When you read the stories you can take the bias into account pretty easily.