The Higgs mass, in the SM, can be renormalized .. but you have to fine tune the parameters to it ... First thing to note is that, unlike the fermion or the gauge boson masses, the Higgs mass is not protected by any symmetry .. An easy way to understand this statement is to remember that, massive gauge bosons or fermions will have one extra degree of freedom (one extra polarizaton) compared to the massless cases. So, one cannot generate extra degrees of freedom just by loop correction. Thus, if a fermion or a gauge boson is massless to start with, it will remain so even after loop correction. But, if a scalar is massless to start with, then no symmetry forbids it to pick up mass via loop correction .. and this can be large.

Why so? Usually the divergent part in the one loop corrections in the fermion or gauge boson masses grows as the logarithm of the Cut-off scale. So even if your cut-off is at 10^15 GeV the divergence is "soft" in the sense that you can absorb it "naturally" into the bare mass and redefine your physical mass.

But, for the 1-loop correction to the Higgs mass, the leading divergence grows as the square of the cut-off. So if you want to get 125 GeV by absorbing this large number into the bare Higgs mass, it will look unnatural. To clarify, suppose you need to get the number 9 by subtracting two numbers and there are two possibilities :

1) 100 -91 = 9 ,

2) 10000000000000000-9999999999999991 = 9 .

Which one would you prefer? Obviously the first one looks more natural, whereas the second option is unnaturally fine-tuned !! Unfortunately, to renormalize the Higgs mass, in the SM, at 125 GeV we have to do the second thing which is aesthetically unpleasant and bothers many people !!!

@Dipankar da nice explanation..