U-Net based algorithms, due to their complex computations, include limitations when they are used in clinical devices. In this paper, we addressed this problem through a novel U-Net based architecture that called fast and accurate U-Net for medical image segmentation task. The proposed fast and accurate U-Net model contains four tuned 2D-convolutional, 2D-transposed convolutional, and batch normalization layers as its main layers. There are four blocks in the encoder-decoder path. The results of our proposed architecture were evaluated using a prepared dataset for head circumference and abdominal circumference segmentation tasks, and a public dataset (HC18-Grand challenge dataset) for fetal head circumference measurement. The proposed fast network significantly improved the processing time in comparison with U-Net, dilated U-Net, R2U-Net, attention U-Net, and MFP U-Net. It took 0.47 seconds for segmenting a fetal abdominal image. In addition, over the prepared dataset using the proposed accurate model, Dice and Jaccard coefficients were 97.62% and 95.43% for fetal head segmentation, 95.07%, and 91.99% for fetal abdominal segmentation. Moreover, we have obtained the Dice and Jaccard coefficients of 97.45% and 95.00% using the public HC18-Grand challenge dataset. Based on the obtained results, we have concluded that a fine-tuned and a simple well-structured model used in clinical devices can outperform complex models.