Many current and future applications for superconducting transition-edge sensor (TES) microcalorimeters require significantly faster pulse response than is currently available. X-ray spectroscopy experiments at next-generation synchrotron light sources need to successfully capture very large fluxes of photons, while detectors at free-electron laser facilities need pulse response fast enough to match repetition rates of the source. Additionally, neutrino endpoint experiments such as HOLMES need enormous statistics, yet are extremely sensitive to pile-up effects that can distort spectra. These issues can be mitigated only by fast rising and falling edges. To address these needs, we have designed high-speed TES detectors with novel geometric enhancements to increase the thermal conductance of pixels suspended on silicon nitride membranes. This paper shows that the thermal conductivity can be precisely engineered to values spanning over an order of magnitude to achieve fast thermal relaxation times tailored to the relevant applications. Using these pixel prototypes, we demonstrate decay time constants faster than 100 μs, while still maintaining spectral resolution of 3 eV FWHM at 1.5 keV. This paper also discusses the trade-offs inherent in reducing the pixel time constant, such as increased bias current leading to degradation in energy resolution, and potential modifications to improve performance.