The learning speed of online sequential extreme learning machine (OS-ELM) algorithms is much higher than that of convolutional neural networks (CNNs) or recurrent neural network (RNNs) on regression and simple classification datasets. However, the general feature extraction of OS-ELM makes it difficult to conveniently and effectively perform classification on some large and complex datasets, e.g., CIFAR. In this paper, we propose a flexible OS-ELM-mixed neural network, termed as fnnmOS-ELM. In this mixed structure, the OS-ELM can replace a part of fully connected layers in CNNs or RNNs. Our framework not only exploits the strong feature representation of CNNs or RNNs, but also performs at a fast speed in terms of classification. Additionally, it avoids the problem of long training time and large parameter size of CNNs or RNNs to some extent. Further, we propose a method for optimizing network performance by splicing OS-ELM after CNN or RNN structures. Iris, IMDb, CIFAR-10, and CIFAR-100 datasets are employed to verify the performance of the fnnmOS-ELM. The relationship between hyper-parameters and the performance of the fnnmOS-ELM is explored, which sheds light on the optimization of network performance. Finally, the experimental results demonstrate that the fnnmOS-ELM has a stronger feature representation and higher classification performance than contemporary methods.