How would one get more quality traffic with this specific challenge: It's a health site where I talk about getting myself well when doctors claim it cannot be done.The first thing to define is "quality traffic". Let's say the health condition is a painful nose wart. If only 14 people in the U.S. have this condition you're never going to get significant traffic. If the condition is indeed widespread you have to ask how people would search for it. Would they type "painful nose warts" or maybe something else like "nose bumps"? If you have been assuming people use the same keyword phrasing you do to describe the same thing you need to investigate what is the actual case. Google's keyword tool can help by showing what people are searching for. (I recommend comparing broad and exact match versions too.) Make a list of all possible phrasing. This can help you identify long tail keywords which there is little competition for, and sometimes strong keywords nobody targets well at all.
After you've confirmed that there is indeed significant traffic to be had for your topic, and you're armed with your keyword list the next step is to set up your site to be successful when aligned with those things. Probably the quickest and easiest thing to do is start a Wordpress blog, since there are many theme possibilities and it's already going to have many SEO basics built in. Next, go to Google and type in: SEO site: news.ycombinator.com. Take a day or two to read through many of the results. Pay extra attention to comments by patio11. Now, consider how you can follow as much of that advice as possible with your site. Be prepared to work and improve on this for days or weeks. The time consuming part of receiving free traffic from search engines really boils down to: creating quality content (keep your phrasing list in mind here!) on the topic, and receiving links to that content. If your content is good enough, links will start appearing naturally. This can take time, but it will happen. To help things along be proactive by reaching out to sites, communities, writers, organizations which have anything at all to do with your topic. Ask them directly for links back, or possibly if you can write a guest post on their site referencing your site. If you follow these steps correctly, over time you will see results.
Now that you've established traffic you need to monetize it. There are a couple ways to do that. If the traffic is significant enough, and depending on the topic, simply putting up ads may be enough. If there are complimentary products to aid in getting healthy, like a special measurement machine, see if you can position yourself to earn a commission whenever it's sold through your site as an affiliate (start contacting suppliers directly if you need to and pitch them).
The other way to monetize your traffic is by selling access to the information itself. You hold valuable information. You want to monetize it. And here I disagree with mechanical_fish. There is nothing wrong with asking people to pay for you taking the time to show them a way to improve their life. Doctors and drug companies don't work free. They are usually quite expensive, and they don't have all the answers as they are literally "practicing" medicine. Providing alternatives which may be more effective and likely less costly as well is certainly worth something.
The approach I would use when taking this route is freely offering about 80-90% of the solution and trying to monetize the remaining 10%. As you say you will probably face skepticism about any proposed solution, whether it's true or not. Talking openly and candidly about the first 90% is a way to address that skepticism. People can detect when you are probably being genuine, and they will respond to that. I would tell my story as genuinely and matter-of-factly as possible on my blog, and interact with readers. Next, I would offer one of two options which could be monetized. My favorite thing to do would be adding a premium membership forum to my site. I'd make it known that in this forum participants shared and progressed together, as well as having access to me. Even if you gave away 100% of your solution, coaching members along is still likely to be a needed and valuable product.
The other option is to package up the remaining 10% in an eBook format where you essentially re-tell your entire story, but with 100% of the information. Show your site visitors that you have taken time to put together a more in depth look at the topic, and you hope they will pay the modest Last, be sure to have a legal disclaimer stipulating that you're not a doctor, you're only telling your story, and people should consult their doctor before beginning any form of treatment for anything.