I think not nearly well enough, but you would never know it from listening to food manufacturers.
Let's start with Europe. Health claims are a big deal there these days, as the agency dealing with them, EFSA (European Food Safety Authority), copes with thousands of petitions.
In June, EFSA representatives complained that the "scientific" evidence submitted by food companies to support their petitions included—get this—"excerpts from the Old Testament, Wikipedia, a Tea Association press release, a Royal Air Force report and the American Heritage Dictionary":
The NDA panel, which is responsible for assessing the mountain of health claims applications submitted under the Regulation, said that along with the expected references to clinical studies published in peer-reviewed academic journals, it had also been presented with references from Wikipedia, press releases, dictionaries, the Bible and even an RAF report.
Trying to make sense of translations of references from other languages into English had presented additional difficulties, while 'clarifications' provided by some applicants in response to requests for further information had been confusing or inadequate, further delaying the process.
The quality of this evidence, says EFSA, is "far from optimal." Indeed.
From the food industry point of view, however, scientific substantiation of health claims presents pesky barriers. Moving on to the United States, a food industry commentator asks:
We're talking about the biggest food companies in the world being told the claims that help sell some of their foods are deceptive and misleading....And them agreeing to change or withdraw the claims ... Er sorry ... but why go along with it if you stand by the science?...Is the science there or not? Do these products (a probiotic drink and an antioxidant-boosted cereal) work or not? ... Does the problem lie with the nutrition science itself (not enough clinically backed, human intervention trial-demonstrated, positive associations), or the way the science is being interpreted by regulators and companies that wish to express some of that science in their marketing materials?
Good questions. As I read the literature, the more carefully done studies of functional ingredients tend to show the least benefit.
As I keep saying, functional foods are about marketing, not health.