2d – How do I procedurally generate rivers on a tilemap with thickness?

I have read numerous articles on the subject but can find none that can explain to me how it is that procedural river generation achieves a certain amount of thickness to the river. I have managed to pice together that I must search for perpendicular vectors to the one I use to construct the river, but am not sure how that is implemented. I am working on a 2D game with tilemaps implemented and don’t know how I iterate through the tiles to achieve the intended result.

I am fairly intermediate when it comes to game development and have just started proc-gen games. Please try to answer simply as to how to do this. I know that others have figured it out and want an answer.

How do I set the thickness of procedurally generated rivers?