Width tool creates unwanted artifacts

Here is a quick tutorial that may help:

Using variable-width strokes

Illustrator will create automatic width points at certain locations if it can't achieve your width creations otherwise.

Please check if there are too many anchor points, perhaps corner points.

Could you please show another screen shot with Width tool and the shape selected so we can see where the width points are? I’m wondering if you have two or more close together before we look for another solution. If so, double-click the one you don’t want and delete it from the pop-up menu.

If that’s not it, we need to look further.

What is your OS and which version of Illustrator? That’s the second place to look.