This happening because you are using Response.write which writes values at the start of HTTP response.
In this case you can follow below steps.
1. Declare protected string variable at class level in your SearchResults.aspx.cs like below
protected string OutPutString ="";
2. Instead of using Response.write, append your string to OutPutString like below
OutPutString += "Family: " + Session["FamilyName"].ToString() + "<br />";
OutPutString +="First: " + Session["FirstName"].ToString() + "<br />";
3. In your SearchResults.aspx display this string where you want to using ASP.NET inline expressions
<%=OutPutString%>
Hope this will help.
One more thing if you are performance concerned then you can use StringBuilder class for appending strings.