Gantt chart not showing correct start dates for all activities

I can't figure out how to separate the data and tell the second list of data to start on the 4th.

Your Gantt chart is a simple stacked bar chart. In order to "separate" data series in a stacked bar chart, you need to add a data series between with the right formatting and separation. Assuming I understood what you are trying to do I:

1) Select column E -> Insert column.
2) In E2 of the new column E, I enter a formula to compute the "duration" for the gap -- the number of days between the end of the week 1 task and the start of week 2 (in column D). =D2-B2-C2. Copy this formula down.
3) Add the new E2:E33 as a new data series to the chart (perhaps use the Select Data dialog).
4) Move the new data series so it is the 3rd data series in plot order. If you use the Select Data dialog, there are up and down arrows just above the "Legend entries (Series)" section of the dialog that can be used to reorder the data series.
5) Format the new series to be "invisible" if desired.
6) Repeat similar procedure for other "gaps".

Can you work with that?