The ever-increasing data rates of modern communication systems lead to severe distortions of the communication signal, imposing great challenges to state-of-the-art signal processing algorithms. In this context, neural network (NN)-based equalizers are a promising concept since they can compensate for impairments introduced by the channel. However, due to the large computational complexity, efficient hardware implementation of NNs is challenging. Especially the backpropagation algorithm, required to adapt the NN's parameters to varying channel conditions, is highly complex, limiting the throughput on resource-constrained devices like field programmable gate arrays (FPGAs). In this work, we present an FPGA architecture of an NN-based equalizer that exploits batch-level parallelism of the convolutional layer to enable a custom mapping scheme of two multiplication to a single digital signal processor (DSP). Our implementation achieves a throughput of up to 20 GBd, which enables the equalization of high-data-rate nonlinear optical fiber channels while providing adaptation capabilities by retraining the NN using backpropagation. As a result, our FPGA implementation outperforms an embedded graphics processing unit (GPU) in terms of throughput by two orders of magnitude. Further, we achieve a higher energy efficiency and throughput as state-of-the-art NN training FPGA implementations. Thus, this work fills the gap of high-throughput NN-based equalization while enabling adaptability by NN training on the edge FPGA.