Help~outlines gets distored if I save my file on PNG with transparency

Hello, it is my first time-saving files on PNG with transparency. I am having trouble with the outlines when it comes to saving with transparency background. I've attached screenshots to make things clear. The first one is with the white background and it appears as it is drawn with nothing being changed. However, the second PNG with transparency background shows that the outlines are being distorted and not as it is drawn. I've tried to draw the outlines with a hard round brush to make sure I didn't miss any spots and also with the pen tool but still appears with the sharp distorted line. Can someone please help me with this problem? Thank you!