Recent advancements in Vision-Language Models (VLMs) have led to the development of Vision-Language Generalists (VLGs) capable of understanding and generating interleaved images and text. Despite these advances, VLGs still struggle to follow user instructions for interleaved text and image generation. To address this issue, we introduce LeafInstruct, the first open-sourced interleaved instruction tuning data with over 30,000 high-quality instances across more than 10 domains. Due to the extensive size of existing VLGs, we opt for parameter-efficient tuning. However, we observe that VLGs tuned with a standard LoRA typically exhibit inferior performance in interleaved text-image generation. We attribute this problem to modality interference and the lack of modality-specialized adaptation design. Hence, we propose Lateralization LoRA, a novel modality-specialized adaptation method inspired by the concept of brain lateralization. Lateralization LoRA employs a hybrid approach, combining the traditional linear LoRA and a Convolutional LoRA for generating text and images, enabling the generation of high-quality text and images by leveraging modality-specific structures and parameter sets. We perform instruction tuning of the VLG (i.e., EMU2) using Lateralization LoRA on the LeafInstruct dataset. Extensive experiments demonstrate that EMU2 tuned with Lateralization LoRA achieve state-of-the-art performance, significantly surpassing baseline models in complex interleaved tasks.