Implicit neural representations and neural rendering have gained increasing attention for bathymetry estimation from sidescan sonar (SSS). These methods incorporate multiple observations of the same place from SSS data to constrain the elevation estimate, converging to a globallynt bathymetric model. However, the quality and precision of the bathymetric estimate are limited by the positioning accuracy of the autonomous underwater vehicle (AUV) equipped with the sonar. The global positioning estimate of the AUV relying on dead reckoning (DR) has an unbounded error due to the absence of a geo-reference system like GPS underwater. To address this challenge, we propose in this article a modern and scalable framework, NeuRSS, for SSS SLAM based on DR and loop closures (LCs) over large timescales, with an elevation prior provided by the bathymetric estimate using neural rendering from SSS. This framework is an iterative procedure that improves localization and bathymetric mapping. Initially, the bathymetry estimated from SSS using the DR estimate, though crude, can provide an important elevation prior in the nonlinear least-squares (NLSs) optimization that estimates the relative pose between two LC vertices in a pose graph. Subsequently, the global pose estimate from the SLAM component improves the positioning estimate of the vehicle, thus improving the bathymetry estimation. We validate our localization and mapping approach on two large surveys collected with a surface vessel and an AUV, respectively. We evaluate their localization results against the ground truth and compare the bathymetry estimation against data collected with multibeam echo sounders (MBESs).