It appears that you have a 7-speed freewheel, not a cassette.
A quick google search found that the TZ37 is a 7-speed freewheel in the Tourney Group.
If this is correct then only a 7-speed freewheel will fit your hub and be compatible with your shifters. And the removal tool must fit a freewheel.
For more speed I suggest that you stay with what you have and turn the cranks faster. With a top gear combination of a 48t chainring and a 14 cog you should be able to maintain at least 27 mph at 100 rpm cadence.