Analysing the original sentence, we have a main clause which contains an extended infinitive construction, which in turn has a depending relative clause.
Es ist immer einfach, {das zu tun, {was die Mehrheit tut.}{rel. clause}}{inf. const.}
The wording of this is in no way perceived as weird, as a number of German speakers have already answered here. Why is that? Well, clearly the ‘was die Mehrheit tut’-relative clause depends only on the ‘das zu tun’ infinitive, so it makes sense having it after that. (German relative clauses typically follow what they are describing.) And of course, German has a tendency of putting infinitive constructions at the very end of a sentence. So this order makes sense, if you consider the relative clause as being part of the infinitve.
What about your suggested reordering?
Es ist immer einfach, {{was die Mehrheit tut,}{rel. clause} zu tun.}{inf. const.}
The main problem here is that relative clauses don’t precede often. They do sometimes, especially if they refer to something rather general:
Was die Mehrheit von mir denkt, interessiert mich doch nicht.
It really just seems so much more natural to have the relative clause follow the infinitive in this case, though. I wouldn’t want to call it ‘wrong’, but if I were marking, I would underline it with a squiggly line to mean ‘not a good way to express it’.
Also note that the relative clause must be flanked by a pair of commas as I added here.