The key to these battles is to try to stay out of Turn for as long as possible, once the enemies are in large numbers, you won't survive running around waiting for your Turn to recharge.
Try to use any attack that you can spam to keep enemies at bay (I know it's random but hopefully you'll get at least one of these): Crash, Spark, Purge, Flood. Even without any upgrades you can use those attacks pretty frequently. I would also suggest buffing your health and defense in your Passive slots (Flood, Tap, Bounce, Crash), and only use Turn when you have Void and high damage attacks (Cull, Flood, Breach) or are in trouble.