Excitable media are extended spatial systems, which support the propagation of waves including pulses and rotating spirals. They are well described by sets of partial differential equations involving a fast activator and a slow inhibitor variable. Here we show that spiral breakup, leading to turbulence, can occur in a two-dimensional reaction-diffusion system with delayed-inhibitor production. Upon a decrease of excitability, spirals become unstable because their wavelengths and periods are too short to be sustained in the system.