You are falling victim to the stateless nature of the pages. As soon as you move to another page, you will loose the state of the first page (i.e. the state that makes the links visible). You will need to save this state in something that is maintained across pages like the session object.
In the onClick handler for the button where you show the links you should also set some session variable to preserve this state:
Session("Page1LinksVisible") = True
In the page_load() for this same page, you should check that session value and show the links if it's true. You'll want to only do this when the page is not in postback mode so it will only preset the page the "first" time it's hit (versus on a postback as in when you click a button).
If Not IsPostBack Then
If Not Session("Page1LinksVisible") Is Nothing Then
links.Visible = CType(Session("Page1LinksVisible"), Boolean)
This is the typical methodology for saving information between pages and between page hits that are not postbacks (postbacks maintain state thru the use of the ViewState).
Work smarter, not harder