To properly think critically you'd have to even question what school teaches you. The framework of school as an institution, it's purposes, it's origins. If you're lucky, your parents teach you how the real world relates to school, how teachers are just normal people, and aren't experts, that schools are operated under a certain ideological agenda, either governmental or from the owners of the private institution. That even experts aren't truth-oracles and have disagreements. That in complicated questions, like history, different countries may teach very different stories in school. And that your school's version is also not unbiased.
I don't see how you can teach the essence of critical thinking when it's in itself a fiercely individualistic don't-just-trust-the-authority idea.
If you teach it as such, you will get people to believe in any and all crackpottery because "I learned not to trust school and experts, I now found the actual truth that my school has repressed in this creationist UFO book on how aliens built the pyramids".
The other option is to teach them not to trust anything that comes from "unapproved" sources, only believe your government institutions, UN orgs etc. This may seem like a good baseline for the average person but it's just appeal to authority and not critical thinking.
I think there is just no such thing as "critical thinking" that could be taught as such, in itself. You have to go to the object level. If you want to dispel creationism, you have to teach biology and talk about how we know what we know about evolution and make sure people deeply understand it in their bones and they don't just regurgitate what they think you expect of them. It's the same in every subject. If you want people not to believe in magic healing crystal energy vibrations and parapsychology and homeopathy, you have to get them to understand some principles of real medicine and real physics (with equations and exercises all that). Only someone who has firm foundations on the object level, can successfully apply critical thinking.
One thing that could be taught though is propaganda techniques, marketing psychology, how it relates to the brain's reward systems, how ads are designed and monitored, A/B testing and tracking in cell phones, addiction. How cults form, the human biases that cult leaders use, a lot of stuff about human behavior, social psychology, trust, different personality types. Fallacies, pitfalls of thinking. But all these are very meta and again, to have a good grasp of these, you need a good actual base on the object level.
Most of actual critical thinking in the real world looks like "wait a minute, that doesn't feel right according to my model of how the world works". It's not really by matching things against a shortlist of logical fallacies that you had to memorize for some test.