The short answer is that your maps aren't that accurate. The ICTZ never really gets beyond ~15N, and at that latitude, its only there for a month or two. Here is a map that might be more accurate.
The ITCZ reverses direction basically as soon as it gets to its northernmost point. You can see this in the climate of N'Djamena, the capital of Chad. ITCZ rolls up that far north (12N in that case) in July, sticks around for about two months, then the place goes back to being a desert. So, while it gets a decent amount of rain for two months, it is also super hot and sunny. That is called the Sahel. So even though some areas of a map may show ITCZ over them, they might only get a month or two of rain; not enough to support savannah grass (though grass will grow in the short rainy season) making them more of a semi-desert scrubland.
Farther north, the rains never really get there at all and you get your giant ergs and wandering Tuareg.
Northern Australia has the ITCZ directly over it, but another factor is at work. It is right on the edge of a warm ocean, so when the ITCZ does show up, there are hot wet winds full of rain. So it ends up getting a lot more rain in the period that the Sahel, and then it gets cyclones as the ITCZ retreats bringing even more rain.