On simple groups of large exponents

It is shown that the set of pairwise non-isomorphic $2$-generated simple groups of exponent $n$ ($n\ge 2^{48}$ and $n$ is odd or divisible by $2^9$) is of cardinality continuum. As a corollary, for any sufficiently large $n$ the set of pairwise non-isomorphic $2$-generated groups of exponent $n$ is of cardinality continuum.