The web and desktop apps display a max of 75 conversations. The web stores all of your conversations, but it is much faster to display only a subset.
If you want to see older conversations, you can archive some of the more recent conversations that you are no longer using. To archive a conversation, simply access it and click the 3-dot menu on the top right, then select Archive Conversation. You can also archive from the Conversations page/Conversation list by clicking the initial/avatar found on the left side of each conversation, then clicking the archive button on the top right.