Is it safe to post one's research on web sites like Math Overflow for "peer review" before submitting to a journal?
If you post under your real name, I think that it is quite "safe" in the sense that your work is unlikely to get stolen because of this.  In fact, posting your work on a public forum is the best defense I know against intellectual theft.

On the other hand, "Is this proof of a known result correct?" is generally not regarded as an appropriate MathOverflow question.  There can be a fine line between asking questions that come up in one's research and asking others to vet your research.  The former is the purpose of the site, whereas the latter is discouraged.  Your question seems to fall on the wrong side.  I think it would stay open, at least, on math.stackexchange.com.

If you have not already published work in the past, I might recommend that you try to get one-on-one help from a qualified (and trusted) mentor, rather than trying to crowd-vet your proof.  It is not impossible to do research mathematics completely independently and with no academic affiliation -- I have seen people do so quite well -- but it is very rare and much harder than doing it with some assistance.  It would be reasonable to seek help from the math department of any university near you.  If the mathematicians you get in contact with can't help you, see if they can connect you to someone who can.

Postscript: I have observed on this site and others that amateur mathematicians often have significant concerns that their work will be stolen.  I think these sentiments are very unfortunate, because they can serve to prevent these people from getting what would benefit them most: qualified help and supervision.  I don't want to say that it is strictly impossible for someone's work to get stolen, but here are two things: (i) I have done mathematics for my entire adult life (more than 20 years), and I have never seen or heard of an amateur mathematician's work being stolen by a professional mathematician.  Reports of theft of mathematical work of anyone by anyone have come to me...but very, very rarely.  (ii) I would estimate that about 90% of the work done by amateur mathematicians is not publishable.  This is not to say that most amateur mathematicians are hopeless or incapable of doing real, publishable work.  However most professional mathematicians train for years and make use of enormous amounts of human capital in order to get to the point where they can do and write up publishable work.  If they had been afraid of showing their work to their mentors, they would not have gotten anywhere in most cases.  Just something to think about.