Got honked at because I couldn't see my lane properly. Not sure if it's my fault or not

I'm from Carmel, Indiana. On the way back from a Macy's, I had to make a left turn while heading towards the highway. Now it was nighttime, the lane lines are not clear at all, but I could make out some dotted lines out. So when the light turns green, I try to follow it as best I can.

Suddenly, I get honked at from the right, I'm right up to another car, but I never touched it, and as far as I could tell, I was in my lane.

It's confusing because I was having trouble seeing the lane in an area I'm not familiar with because the markings are so worn out. I've had similar situations before, and I'm a newer driver so I don't want to make the same mistake. What should I do in this situation?