1. Arjuna defeated everyone except Bhishma in the Virat war, single handed, with his inexperienced charioteer (Bhishma was not affected by Arjuna's "Sammohana" weapon while everyone else were made senseless by that weapon. But acts as if fainted in order to make that battle end). This proves not only Arjuna's powers but also Bhishma's powers.
2. When the plan of capturing Yudistra was plotted, Drona told Duryodana that he would not be able to capture or even go near to Yudistra if Arjuna was there (So, Arjuna was challenged by Susarman and taken to a far away battle field to aid Drona's mission of capturing the King). One day when Drona was about to capture Yudistra, Arjuna came into the scene and made Drona withdraw (When Duryodan balms Drona for failing to capture Yudistra, Drona replied "I told you already, I can't capture Yudistra before Arjunas powers"). Drona always considered Arjuna as powerful than him.
3. Arjuna was extremely devoted to Drona and he always try to fight defensive battles only with Drona as he does not want to hurt him. (It has been repeated several times in Vyasa bharata's original translation by KMG). Once he was asked to kill Drona, Arjuna says "I'd kill myself instead of even thinking about killing my guru"
4. On the day Arjuna killed Jayatrata, Arjuna transgressed Drona (and Karna) and when Duryodana asked Drona, Drona told him "I could not stop Arjuna as Arjuna's shooting length and speed are higher than of mine"
5. Arjuna defeated Karna in Virat war, Drowpathi suyamvar (when Karna went to attack Drupata, Arjuna fought with Karna, broke Karna's bow and made Karna withdraw) and also Karna fled in war with Gandarvas whereas Arjuna was the one won that battle. Karna never defeated Arjuna according to the original Mahabharata. (just Karna relied on his 'Shakthi' weapon to kill Arjuna as he knew he can't kill Arjuna with his skills but, Arjuna too had Pasupatha weapon that could simply kill Karna, but Arjuna never relied on divine weapons to win over his arch enemy.)
6. When rating all warriors right before the war, Bhishma rated everyone (Ratha, athi-ratha, maha-ratha, etc and at the end tells that Arjuna's the foremost of all warriors taking part in that battle. (This shows that Arjuna was the most powerful warrior there but we can't use this statement to say he's better than Bhishma since Bhishma was humble enough to not rate himself. Also, Arjuna obviously failed to throw Bhishma off though he tried hard). Drona also acknowledged Bhishma's statement (Drona's statement comes as a reply to Karna when Karna interrupted Bhishma during the rating process. Drona also reminds the Virat war defeat and the battle with Gantharvas to Karna - In KM Ganguly's translation)
7. Karna was defeated by Bhima and Sathyaki in some of their duels in the final Mahabharata battle. But, Bhima or Sathyaki could never defeat Bhishma. They ran away after some time whenever they fought with Bhishma.
8. Abhimanyu made Karna and Drona fly away on the day he penetrated Chakravyuga. They could not stop the aggressive Abhimanyu (Karna broke his bow from behind as he could not face him from front). But Abhimanyu could not defeat Bhishma in any duels they encountered during the first ten days of the battle.
9. Drona considered Bhishma as more powerful than him and even no one else believed that Drona was above Bhishma. Moreover, Bhishma vs Arjuna and Drona vs Arjuna encounters show that Bhishma was powerful than Drona
10. Arjuna Killed Karna's son Vrishasena in a direct encounter before Karna's eyes as a revenge to Karna's contribution to Abhimanyu's killing by unfair means. And Karna could not overpower Arjuna and protect his son.
Most of the above points can not be found in any Mahabharata TV serials or the commonly available (abridged) Mahabharata books.
Incidents 1,2,3,4,6 proves that Arjuna was powerful than Drona.
Incidents 1,4,5,6,10 proves Arjuna was powerful than Karna.
Incidents 1, 8, 9 proves that Bhishma was powerful than Drona.
Incidents 1, 7, 8 proves that Bhishma was powerful than Karna.
Bhishma vs Arjuna who is better?
Arjuna never defeated Bhishma (in a fair fight) and Bhishma never defeated Arjuna either. Also, none of the other great warriors of Mahabharata defeated Arjuna or Bhishma. (This is according the original phrases of Mahabharata).
It's a great argument (As long as right facts are presented) since, still there are no straight forward evidences to show Arjuna's superiority over Bhishma or Bhishma's superiority over Arjuna. They failed to defeat each other in all the combats they encountered.
Karna and Drona were great warriors and close enough to Bhishma or Arjuna but, not better than or equal to Arjuna or Bhishma.
On Karna Vs Arjuna issues...
For the people who did not read the long unabridged Mahabharata, Karna would look more powerful than what he actually was in the original Mahabharata. It is the present day writers and film makers made Karna look bit more powerful because Karna's character is more appealing from the present day point of view (he can be easily made as a modern day hero) so, he has been often exaggerated. Of course Karna was a great warrior and almost equal to Arjuna but, his powers were bit exaggerated today due to the mentioned reason. I too believed that Karna was bit powerful than Arjuna until I read and researched the unabridged Mahabharata text.
On Barbarika's story
Barbarika's story came via a folk tradition in some parts of North India. The original unabridged Mahabharata text does not mention any such story.
Sources for the facts mentioned above
1. The Mahabharata of Krishna-Dwaipayana Vyasa translated from Sanskrit to English by Kisari Moham Ganguly - Online archive available at
2. Mahabharata by C. Rajagopalachari (Available as volumes in archive.org)
(Those two are the real fully unabridged Mahabharata translations available today)