Issue is because of
AutoPostBack=true.
Because of this your page is get reloaded.
So instead of that you can achieve the same functionality using javascript without page postback as mentioned below :
TextBoxes :
<asp:TextBox runat="server" ID="TextBox1" onchange="ManageText(this.value)"></asp:TextBox>
<asp:TextBox runat="server" ID="TextBox2"></asp:TextBox>
JavaScript Method :
function ManageText(value)
{
document.getElementById('<%=TextBox2.ClientID%>').value = value;
}