<asp:UpdatePanel runat="server">
        <ContentTemplate>
            <asp:Button Text="text" runat="server" ID="btn1"/>
            <asp:TextBox ID="TextBox1" runat="server"></asp:TextBox>
        </ContentTemplate>
    </asp:UpdatePanel>
    <asp:Button Text="text" runat="server" ID="btn2"/>
    <asp:TextBox ID="TextBox2" runat="server"></asp:TextBox>

protected void Page_Load(object sender, EventArgs e)
    {
        TextBox1.Text = TextBox2.Text;
    }

<asp:UpdatePanel runat="server" UpdateMode="Conditional">
    <Triggers>
        <asp:AsyncPostBackTrigger ControlID="btn1" />
    </Triggers>
    <ContentTemplate>
        <asp:Button Text="text" runat="server" ID="btn1"/>
        <asp:TextBox ID="TextBox1" runat="server"></asp:TextBox>
    </ContentTemplate>
</asp:UpdatePanel>
<asp:Button Text="text" runat="server" ID="btn2"/>
<asp:TextBox ID="TextBox2" runat="server"></asp:TextBox>