What you're calling curves are splines, and I don't believe that LazyCAM supports splines.
Since there are no g-codes for splines, they'll eventually get converted to straight lines or arcs.
In AutoCAD, you can try drawing with arcs instead of splines.
For you SVG, you'd need to use something that can fit arcs to your curves.
Better yet, use something better than LazyCAM.