Topological photonic states, inspired by robust chiral edge states in topological insulators, have recently been demonstrated in a few photonic systems, including an array of coupled on-chip ring resonators at communication wavelengths. However, the intrinsic difference between electrons and photons determines that the 'topological protection' in time-reversal-invariant photonic systems does not share the same robustness as its counterpart in electronic topological insulators. Here in a designer surface plasmon platform consisting of tunable metallic sub-wavelength structures, we construct photonic topological edge states and probe their robustness against a variety of defect classes, including some common time-reversal-invariant photonic defects that can break the topological protection, but do not exist in electronic topological insulators. This is also an experimental realization of anomalous Floquet topological edge states, whose topological phase cannot be predicted by the usual Chern number topological invariants.