Seeing a battle between Superman and Captain Marvel would be interesting but we have deduced that the Last Son of Krypton will be taken down by the strongest hero of Marvel, it could be a little different for Supergirl if she were to go up against Captain Marvel. It has been proven on many occasions that Supergirl is even stronger than Superman so Captain Marvel is going to have a tough time taking her on.
The last survivor of Krypton after her cousin Kal-El was Kara Zor-El. She was sent to earth as the protector of Kal-El, but her pod got knocked off course, so she arrived later on Earth, till the time Kal had already dwelled and become Superman. Her alter ego on Earth is Kara Danvers and just like her cousin, she is a reporter Journalist at CatCo in National City.
Being a Kryptonian, under the yellow sun she possesses Super-Strength, Super-Speed and agility, heat vision, freeze breath, X-ray vision and the ability to fly at speeds much faster than a speeding bullet. She is actually the fastest Kryptonian alive as she is much faster and even stronger than Kal El if her full strength is achieved. Having lived on Krypton for a while too, she has a range of abilities much wider than Superman himself even though she does share the same weaknesses against Kryptonite, Red Sun Radiation, and magic.
One thing people don’t know about her is that she also joined the Red-Lantern Corps and became a Red Lantern herself. So, just imagine a being as strong as her having an upgraded power level with the Red Power Ring. Moreover, the Red Ring of Rage unlocks even more powers for her as she gets the acid spit, ability to create wormholes and anything she wants with the ring. Her power gets a boost with the rage of the Red-Lantern.
Other, than that, she has been able to turn her freeze breath into sonic screams at times. In the comics at times, she has shown Pyrokinesis, Telekinesis, invisibility, and has telepathic invulnerability. If she unlocks every bit of her strength, she will be much powerful than Superman and totally unbeatable by anyone.
Carol Danvers aka Captain Marvel entered the air-force to portray her life-long dream of flying. She was later recruited as an intelligence agent for CIA because of her stellar performance, superb combat skills and natural intellect and that is where she met Nick Fury. Later she even took a position with NASA and from here on her mortal enemy Col. Yon Rogg kidnapped her and used her as bait against Mar-Vell as the two had a history of conflicts.
So, here Carol was caught in the middle of a fierce battle, and along with Mar-Vell, she was present in an explosion of a Kree Psyche-Magneton device. This is where Danvers became Captain Marvel as the energy bombardment somehow caused Carol’s genetic structure to be melded with Mar-Vell’s and Earth got a perfect hybrid of Kree and human genes in addition with Mar-vell’s Kree knowledge, training and a massive set of powers.
Captain Marvel is insanely powerful as she possesses much more than the cliché Super Strength, Speed, Stamina, Flight, and Durability. Along with these basic powers, she has a sixth sense which is a precognitive ability that allows her to subconsciously recognize danger even before it occurs, and even predict an enemy’s attacks even before they make them.
Added to this, she has enormously strong energy blasts, moreover, she can even absorb energy and become even stronger with her attacks. She can survive in space and vacuum, defy and control gravity around others as well!
But the power that actually makes her the strongest of all is that she transforms into her ultimate form which is called Binary. This allows her to reach her peak strength and go even beyond that. The level of her strength in Binary cannot be calculated practically and all her attacks and abilities that she possesses become even stronger. She even gains a wider range of abilities too. In her Binary form, her strength and flying ability get amplified which allows her to go even faster than the light-speed!
Supergirl vs Captain Marvel
A battle between these two will have to take place on an uninhabited planet or in the cosmos as they will literally destroy everything around them in pursuit of taking each other down. Both of them have the strength to level planets. Supergirl would fight similar to Superman in most cases so her powers again may not be of any big effect against Captain Marvel. But some of her powers like the sonic scream and telekinesis in some cases could cause Carol some real trouble.
The real battle will begin when Captain Marvel goes Binary and Supergirl uses her Red Ring Rage against Captain Marvel. It is then that the battle would get really even as Supergirl would get access to a set of whole new powers which Captain Marvel might find difficult to counter.
Still, it is safe to say that Captain Marvel in her Binary would still be much stronger and she will ultimately emerge Victoriously. But if Supergirl were to take help from her Kryptonian Brother, then the two would have the upper hand no matter how strong Captain Marvel gets in her Binary form!