Trim path is animating in a weird way

It isn't the Convert to Shape command that is the problem, the Illustrator path probably wasn't created in an efficient way that would make it animatable in AE.

Very often shapes that are created in Illustrator are not created in a way that would make them easy to animate in AE.

For example, in your screen shot the converted shape layer is made up off 5 different paths, each of them is being animated as part of the trim path effect. 

For a shape animmation like this you want one path, the one that creates the visible shape you want to animate.

It's possible that the extra paths are just junk that was created in Illustrator and can just be deleted, or they could be the result of the shape being drawn badly in Illustrator (separate paths being placed next to each other so they look like a single line). Without a bit of trail and error there is no way to know.

Based on the way it is animating it looks to me like the visible design was created in Illustrator by placing multiple individual paths next to each other. This would work for illustrator purposes but (since it ony cares about the visual appearance and not the structure) when trying ot animated in AE it causes a problem.

If they are multiple paths you could fix it in Illustrator by using the Join command (Object > Path > Join) or just redraw it as a single path.

make sure that your shape doesnt has a white stroke that we couldnt see over the white background

Make sure that there is only one shape in the shape layer.

The auto-conversion often makes crazy stuff.

*Martin