Is 7-speed for 10-speed cassette replacement possible?
Higher number of speed cassettes and freewheel are not wider, they have smaller sprocket spacing.

If you were talking about a cassette/freehub system, you would be able to fit a 10 speed cassette on the existing hub because the freehub body is the same width for 7 to 10 speed cassettes. (11 speed requires a wider freehub body on road bikes).

I'm not terribly familiar with freewheels, but I would think the same principle holds.

However, if you have a 7 speed rear, freewheel equipped, Tourney level bike, and you want a 10 speed rear, you really want a whole different bike. To get a 10 speed rear you need to upgrade to Tiagra or Deore, depending on if you have a road or mountain style bike. You'll need to replace the crankset as well as the derailleurs and shifters. Buying the bare drivetrain groupset will be expensive, require a lot of time to install, require special tools and will give you a lot of headaches trying to get it set up correctly. 

Go the easy route and get an upgraded bike. It will probably be cheaper and you'll get an upgrade on every component, not just the drivetrain and it will work right out of the box.