Why are some MLE's hard to compute?

Numerical procedure for finding MLE for Normal family converges fairly
fast (ie, O(n) computer operations to get n bits), whereas similar
procedure for finding MLE for Gamma family converges slowly (O(?)). Is
there something useful that can be said about when MLE estimation is
difficult in exponential families?

Does it have something to do with Fisher Information matrix being
diagonal everywhere perhaps?

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