Since many applications of terahertz spectroscopy depend on spectral bandwidth, the generation of ultra-broadband terahertz radiation is one of the most important challenges faced by terahertz scientists. Spintronic terahertz emitters (STEs) are promising sources to produce very compact terahertz pulses and broadband spectra. Here we optimized the iron/gold heterostructure based on the metal layer thicknesses for the first time. The optimization was performed experimentally based on the obtained theoretical results. The optimized thicknesses were obtained as 2 and 5 nm for Fe and Au layers, respectively. The experimental results were in very good agreement with corresponding theoretical results. The optimized STE generates terahertz bandwidth up to 4 THz which is limited by the frequency response of the used detector. The dynamic range is well above 60 dB with the maximum around 0.5 THz. The results show that the presented STE is capable of producing large amplitude terahertz pulses and broadband spectral range. Based on our knowledge, this is the first time that the Fe/Au spintronic terahertz emitter has been optimized.