1. Navigate to <your Steam folder>/steamui/css
2. Open sp.css
3. Search for libraryhome_UpdatesContainer_17uEB
4. For the first two occurences, replace height:324px and height:300px respectively (within the curly braces after them) with display:none.
5. Restart Steam

Notes:

• Adding display:none to the first rule already worked for me, but might not for everyone.
• You have to make sure you don’t change the length of the file or Steam will redownload it on next launch. Replacing the height attribute is just very convenient, because it has the exact same length as display:none.

Edit for the Linux people:

perl -pi -e 's#(\.libraryhome_UpdatesContainer_17uEB\{[^\}]*?)height:[0-9]{3}px#\1display:none#g' ~/.steam/steam/steamui/css/sp.css