The scientific method is hard to explain, and scientists frequently mention peer reviewed papers when talking to the public. I think that this is mainly to dismiss things like health anecdotes that spread around the internet. We don't have time to explain in detail why your uncle's implausible and untested miracle cure for whatever disease is unlikely to work.

There's no easy answer to which sources are reliable, but it helps to look at what reputable organizations say. They typically base their advice on the majority opinions of relevant experts. Scientists are human, so they are bound to make mistakes, but in the whole they have made demonstrable progress in a variety of fields.

Peer reviewing is mostly orthogonal to journal publishing. Publishing in a journal generally requires a peer review, but there's no reason why you need a journal publication to get a peer review. Journals largely exist to manage prestige and career advancement in academia.

As for what science lay people can rely on? I'm not sure there's a simple answer to that question.

I think you either have to find scientists you can trust (which presents a chicken & egg process) or study the field yourself.

That said, there are some good heuristics. Good scientists are able to show their reasoning process and explain why X is bad/good science. Bad ones handwave everything and push credentialism or rely on fallacious reasoning.

For example, I remember long ago when Ars Technica posted a detailed explanation of both what homeopathy is and how we know that it's completely bogus, why "water memory" doesn't and cannot exist, etc.

Meanwhile, in another failure of peer review, Nature published nonsense on "water memory" back in 1988:

https://en.wikipedia.org/wiki/Water_memory