Morphogenesis often yields organs with robust size and shapes, whereas cell growth and deformation feature significant spatiotemporal variability. Here, we investigate whether tissue responses to mechanical signals contribute to resolve this apparent paradox. We built a model of growing tissue made of fiber-like material, which may account for the cytoskeleton, polar cell-cell adhesion, or the extracellular matrix in animals and for the cell wall in plants. We considered the synthesis and remodeling of this material, as well as the modulation of synthesis by isotropic and anisotropic response to mechanical stress. Formally, our model describes an expanding, mechanoresponsive, nematic, and active fluid. We show that mechanical responses buffer localized perturbations, with two possible regimes-hyporesponsive and hyperresponsive-and the transition between the two corresponds to a minimum value of the relaxation time. Whereas robustness of shapes suggests that growth fluctuations are confined to small scales, our model yields growth fluctuations that have long-range correlations. This indicates that growth fluctuations are a significant source of heterogeneity in development. Nevertheless, we find that mechanical responses may dampen such fluctuations, with a specific magnitude of anisotropic response that minimizes heterogeneity of tissue contours. We finally discuss how our predictions might apply to the development of plants and animals. Altogether, our results call for the systematic quantification of fluctuations in growing tissues.