TTG is currently working on a patch for the download problems.
If you already tried connecting through a fast (as in >=50MBit/s) and stable connection, while leaving the App running in the foreground, waiting for the patch is probably the least hassle.
One user also was able to prevent disconnections, by
touching the download screen about every 30 seconds. I don't know much about this "ritual", but if it does help, longer intervals could suffice.
If you want to try a workaround, instead of waiting for the patch, you can try if connecting through a VPN helps. There are free VPN apps in the iTunes store, one example is
Hotspot Shield.