Neural implicit scene representations have recently shown encouraging results in dense visual SLAM. However, existing methods produce low-quality scene reconstruction and low-accuracy localization performance when scaling up to large indoor scenes and long sequences. These limitations are mainly due to their single, global radiance field with finite capacity, which does not adapt to large scenarios. Their end-to-end pose networks are also not robust enough with the growth of cumulative errors in large scenes. To this end, we present PLGSLAM, a neural visual SLAM system which performs high-fidelity surface reconstruction and robust camera tracking in real time. To handle large-scale indoor scenes, PLGSLAM proposes a progressive scene representation method which dynamically allocates new local scene representation trained with frames within a local sliding window. This allows us to scale up to larger indoor scenes and improves robustness (even under pose drifts). In local scene representation, PLGSLAM utilizes tri-planes for local high-frequency features. We also incorporate multi-layer perceptron (MLP) networks for the low-frequency feature, smoothness, and scene completion in unobserved areas. Moreover, we propose local-to-global bundle adjustment method with a global keyframe database to address the increased pose drifts on long sequences. Experimental results demonstrate that PLGSLAM achieves state-of-the-art scene reconstruction results and tracking performance across various datasets and scenarios (both in small and large-scale indoor environments). The code will be open-sourced upon paper acceptance.