Ah! You linked a kurzgesagt video I see, its definitely one of my favorite channels on Youtube.
Anyways, at the pain of sounding pessimistic, I can say it will be nearly impossible to defeat big oil any time soon. Oil tycoons in Saudi Arabia have almost as much as power in the U.S. government as senators. Economy and Politics have always been intertwined throughout history, so those who own economic power, also hold political power to an extent. This can be seen in the corporate lobbying so frequent in congress. The only way to start replacing oil with clean energy would be through political legislation, and, as mentioned, since Oil has so much power in our government, any law to do so would likely be rejected by congress. So, you may ask, does this mean that our political system is practically broken because it must abide by uncountable private power and can not do what it must for the greater good? I'll let you decide.
P.S.: That's an interesting thing Kraichgauer mentioned about nuclear power being an enemy to big oil. I wouldn't be surprised if many of the horror stories about nuclear energy were exaggerated and spread around the media with a little push by Oil companies. Chernobyl must have been a godsend for them.