Careful: democracy is not about majority, but rather (literally) "government by the people"; there is quite a difference. There are various types of "majority", and all of them fail at fully representing "the people" as a whole, unless it's a 100% majority where nobody abstains (and even then, one would have to argue about who was allowed to vote and how).
This is why any robust democratic system will have, on some subjects, "qualified" majorities (where the threshold is set much higher than 51%, or abstained votes are counted against, etc etc) or "relative" majorities (where the most popular option wins, even when it doesn't represent at least 50% of the votes) as well as the "simple" majority (usually 50% + 1 or 51% of the expressed votes). They're different ways of really interpreting "the will of the people" on different subjects. A proper democratic system recognises the importance of respecting minorities and the volatility (and gullibility) of popular sentiment, and safeguards itself against wrong decisions. Many fascist governments were democratically elected and survived re-election, but they were not democratic in nature.
If the majority of synapses in your brain should decide tomorrow that shooting yourself in the foot is a good idea, despite "a small but vocal minority opposed to the plan" (i.e. your foot), that doesn't mean that it'd be in your real interest to do so.