When in the Web Reader click of tap in the center of the screen to bring up the Menu. In the top left the Arrow will let you exit the Manga/Comic and the Title, Volume, and Chapter of what you have open is displayed.
Note: All Menu options scale with the screen size of your device display.
Warning: Web Reader menu settings do not persist between reading sessions. Use User Preferences to update for persistence.
The reader provides a few ways to read without having to close the reader to open the next chapter. As shown above, there are Next/Previous Chapter buttons that can be manually clicked and in addition, for non-webtoon mode, you can just page like normal and within 10 pages of a chapter boundary, the server will prefetch information needed and tell the server to start caching said files.
For Webtoon mode, due to the nature of the reader, there is a slightly more manual way to trigger the next/previous chapters. You can always use the buttons, however to do it without using the menu is more ideal. Now when the user scrolls to top or last page in the reader, a "spacer" is inserted at the top or bottom respectively. This spacer should show an animation to the user to scroll into it. When that is done, the next chapter is loaded.