You can use javascript in order to achieve that, add the "onChange" attribute to the text box and when the text box value changes then directly change the label text.
Put this in Page_Load event in code behind:
this.tbDrugName.Attributes.Add("OnChange", "tbDrugName_OnChange()");
And add this javascript function in markup:
function tbDrugName_OnChange() {
var txtVal = document.getElementById('<%= tbDrugName.ClientID %>').value;
document.getElementById('<%= lblList.ClientID %>').innerText = txtVal;
}