The rate of strong convergence is investigated for an approximation scheme for a class of stochastic differential equations driven by a time-changed Brownian motion, where the random time changes $(E_t)_{t\ge 0}$ considered include the inverses of stable and tempered stable subordinators as well as their mixtures. Unlike those in the work of Jum and Kobayashi (2016), the coefficients of the stochastic differential equations discussed in this paper depend on the regular time variable $t$ rather than the time change $E_t$. This alteration makes it difficult to apply the method used in that paper. To overcome this difficulty, we utilize a Gronwall-type inequality involving a stochastic driver to control the moment of the error process. Moreover, in order to guarantee that an ultimately derived error bound is finite, we establish a useful criterion for the existence of exponential moments of powers of the random time change.