Oil leaks from the front of the tanker, but oil tankers do not store oil in this way.
When the news is showing that "Supergirl saves couple trapped in car," underneath the text it says "Family of the couple incredible thankful" instead of "incredibly thankful." Also right after this on another news station, they use "it's" in the scrolling text at the bottom of the screen when it should say "its"
Supergirl is frequently seen wearing earrings. Given that she is virtually invulnerable and heals rapidly this would be extremely difficult to achieve.