Transformers have achieved tremendous success in various computer vision tasks. By borrowing design concepts from transformers, many studies revolutionized CNNs and showed remarkable results. This paper falls in this line of studies. More specifically, we introduce a convolutional neural network architecture named ParCNetV2, which extends position-aware circular convolution (ParCNet) with oversized convolutions and strengthens attention through bifurcate gate units. The oversized convolution utilizes a kernel with $2\times$ the input size to model long-range dependencies through a global receptive field. Simultaneously, it achieves implicit positional encoding by removing the shift-invariant property from convolutional kernels, i.e., the effective kernels at different spatial locations are different when the kernel size is twice as large as the input size. The bifurcate gate unit implements an attention mechanism similar to self-attention in transformers. It splits the input into two branches, one serves as feature transformation while the other serves as attention weights. The attention is applied through element-wise multiplication of the two branches. Besides, we introduce a unified local-global convolution block to unify the design of the early and late stage convolutional blocks. Extensive experiments demonstrate that our method outperforms other pure convolutional neural networks as well as neural networks hybridizing CNNs and transformers.