A lot of turners used to use faceplates to turn bowls. The faceplate was mounted and the the outside of the bowl was formed. Then the inside was hollowed out. I have couple of bowls from the 20's and 30's that my grandmother used that grandpa made. When he took the faceplate off he filled the screw holes with sanding dust and glue mix. It can be done. You can buy some pretty good quality chucks that don't break the bank. It would help if we knew the lathe you have and I am guessing it is 1 X 8 spindle. Grizzly has one that I have 5 of for about $110 each. What gouge are you using on your bowls? Is it a spindle gouge or a bowl gouge?
To succeed in life, you need three things: a wishbone, a backbone and a funnybone.