Diffusion-weighted magnetic resonance imaging (DW-MRI) of fetal brain is challenged by frequent fetal motion and signal to noise ratio that is much lower than non-fetal imaging. As a result, accurate and robust parameter estimation in fetal DW-MRI remains an open problem. Recently, deep learning techniques have been successfully used for DW-MRI parameter estimation in non-fetal subjects. However, none of those prior works has addressed the fetal brain because obtaining reliable fetal training data is challenging. To address this problem, in this work we propose a novel methodology that utilizes fetal scans as well as scans from prematurely-born infants. High-quality newborn scans are used to estimate accurate maps of the parameter of interest. These parameter maps are then used to generate DW-MRI data that match the measurement scheme and noise distribution that are characteristic of fetal data. In order to demonstrate the effectiveness and reliability of the proposed data generation pipeline, we used the generated data to train a convolutional neural network (CNN) to estimate color fractional anisotropy (CFA). We evaluated the trained CNN on independent sets of fetal data in terms of reconstruction accuracy, precision, and expert assessment of reconstruction quality. Results showed significantly lower reconstruction error (n=100,p<0.001) and higher reconstruction precision (n=20,p<0.001) for the proposed machine learning pipeline compared with standard estimation methods. Expert assessments on 20 fetal test scans showed significantly better overall reconstruction quality (p<0.001) and more accurate reconstruction of 11 regions of interest (p<0.001) with the proposed method.