You have to consider compatibility between the shifter/brake lever units and:
- Rear derailleur
- Front derailleur
- Brake caliper
Rear derailleur should be fine, Shimano uses the same cable pull ratio for everything 9 speed an below.
Front derailleur is probably OK, although I'm not familiar with Shimano's schemes for front derailleur cable pull ratio.
Brakes can be problematic is there is generally a difference between the leverage ratio in the levers and calipers for 'road' style U-brakes and MTB V-brakes.
Even if you can work out the compatibility issues, the cost of replacing the bars and shifters can be significant compared to the purchase cost of the bike (especially at the Tourney level). Sometimes selling the bike and purchasing one more suitable for your needs is the better way to go.