I don't think you understand what the difference really is between a TextBox and a ListBox. Personally, I can see why you wouldn't want a ListBox, but memory management is in no way the same. If you want to see how it's really done, I suggest getting the .NET reflector and looking at it.
A TextBox doesn't store the text as an array of strings representing the lines. When you call .Lines[xxx], the TextBox first builds the array of strings representing each line. Then, it passes the whole array and you select the one you want.
Though AppendText is similar to the ListBox.Items.Add. However, the line deletion will be less efficient.
Anyway, how about this:
output02.Text = output02.Text.Remove(0,output02.GetFirstCharIndexFromLine(5));
Also, just some hints with scrolling and flickering. If you want to pause the redrawing of the textBox so that the user can't see the scroll changing, you can use the code found at:
Preventing controls from redrawing[
^]
It's in VB, but you should be able to convert it easily. It will make it look better.