The design problem of a high-gain observer is considered for some 2 × 2 and 3 × 3 semilinear reaction-diffusion systems, with possibly distinct diffusivities, and considering distributed measurement of part of the state. Due to limitations imposed by the parabolic operator, for the design of such an observer, an infinite-dimensional state transformation is first applied to map the system into a more suitable set of partial differential equations. The observer is then proposed including output correction terms and also spatial derivatives of the output of order depending on the number of distinct diffusivities. It ensures arbitrarily fast state estimation in the sup-norm. The result is illustrated with a simulated example of a Lotka-Volterra system.