Hi iammann,
You need to mention in sweetalert option "html:true" so that sweetalert will come to know that there is HTML content used in it and there is no need of using single apostrophe while using HTML tags so below is the right way of using it.
Refer below code
string Message = "Your Support Reference No. is : <b> " + Convert.ToString(dt.Rows[0]["flag"]) + " </b> which has been sent on your email. Kindly check the status of your request within next 5 days.";
string str_alert_Msg = @"swal({
type: 'success',
title:'Good job!',
text: '" + Message + @"',
html: true
});";
ScriptManager.RegisterStartupScript(this, GetType(), "Success", str_alert_Msg, true);