Your second timer is silly - get rid of it.
The following should solve your problem ...
a) Make sure that textbox1 is defined as Multiline
b) Ensure that the textbox1 is big enough or more sensibly has ScrollBars = True
c) Add a System.Environment.NewLine to TweetText before adding it to the textbox (to make sure each tweet is on a separate line, not disappearing off to the right of the textbox)
d) Position the cursor at the bottom of the text box each time (if it scrolls or there is too much data for the size)
TextBox1.Text += (TweetText) + System.Environment.NewLine
textBox1.SelectionStart = textBox1.Text.Length
textBox1.ScrollToCaret()
[EDIT - see OP comments, still not working]
Try putting a breakpoint on
For Each Tweet As String In Tweets
When your code breaks hover over
Tweets
and check the bit that says "Count=x" - make sure there is something in the List.
If it says "Count=0" then you will need to examine your function
parseTweets()
If it says "Count=1" (or more) then step onto the next line and have a look at TweetText ... if that is blank then it could be your regex that's at fault