I suppose it depends on how you define "big." I've thought about this a lot actually(without losing any sleep over it)and I'd say Genesis were about as big in the seventies as Rush were(more or less)or as big as REM was in the 80's.
They weren't as big as Pink Floyd, Yes or ELP in the period of 71-76 or so but they were still probably household names in college and prog circles. In ENgland, Italy and Europe in general they were just as big if not bigger than those bands(although starting out initally maybe not quite as much since they did open up for VDGG in 71 in Italy (and probably elsewhere)and not the other way around. But even in the US I'd say Genesis were still fairly well known even before Steve Hackett left and I doubt there were many hardcore prog fans who didn't at least know who they were by 1974/75.
The unfortunate truth is that Genesis didn't get real big until after Steve left(at least in the US). Their first gold album was "and then there were three" which many people see as the beginning of their pop period(I see it(along with Duke)as being more of a transitional album though). Trick of the Tail was their first top forty(with Lamb just missing it at number 41 but apparently it was on the charts a while). Also, "wind and wuthering" was the first time the band played at the Spectrum in Philly but I suppose that doesn't count since the OP was wondering about their 70-75 period.
Also, it's worth mentioning that afaik Genesis didn't get on a major label until 73 with Atlantic and the SEBTP album. That is probably what landed them in the US charts for the first time at a not too shabby number 70(same spot as PF's Meddle about two years earlier).
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