We find a breakdown of the critical dynamic scaling in the coarsening dynamics of an antiferromagnetic {\em XY} model on the kagome lattice when the system is quenched from disordered states into the Kosterlitz-Thouless ({\em KT}) phases at low temperatures. There exist multiple growing length scales: the length scales of the average separation between fractional vortices are found to be {\em not} proportional to the length scales of the quasi-ordered domains. They are instead related through a nontrivial power-law relation. The length scale of the quasi-ordered domains (as determined from optimal collapse of the correlation functions for the order parameter $\exp[3 i \theta (r)]$) does not follow a simple power law growth but exhibits an anomalous growth with time-dependent effective growth exponent. The breakdown of the critical dynamic scaling is accompanied by unusual relaxation dynamics in the decay of fractional ($3\theta$) vortices, where the decay of the vortex numbers is characterized by an exponential function of logarithmic powers in time. ; Comment: 13 pages, 26 figures