  In this paper, we study the transition densities of pure-jump symmetric
Markov processes in $ {{\mathbb R}}^d$, whose jumping kernels are comparable to
radially symmetric functions with mixed polynomial growths. Under some mild
assumptions on their scale functions, we establish sharp two-sided estimates of
transition densities (heat kernel estimates) for such processes. This is the
first study on global heat kernel estimates of jump processes (including
non-L\'evy processes) whose weak scaling index is not necessarily strictly less
than 2. As an application, we proved that the finite second moment condition on
such symmetric Markov process is equivalent to the Khintchine-type law of
iterated logarithm at the infinity.
