You may have mislaid a piece when you took your Bissell apart to clean it after using it the first time.
I experimented with mine to try and reproduce your issue. On the bottom of the water tank, there are two pieces that can be removed from it. One of them looks like a black plug at one end, with a red rubber flange piece on the other end. If you push your finger inside the red rubber piece, you can feel a spring move up and down. This is the first piece you place on the bottom of the water tank. I suspect you have this piece, because if you didn't, water would gush out of the water tank as soon as you turned it over after filling it.
The other piece, which is the one I think you may have have mislaid, is a larger black, plastic ring with screw threads. This piece goes over the top of the first plug piece, screws onto the water tank, and holds the plug in place. The words "RECLOSE TIGHTLY" appear on mine. If this piece is missing, the water tank does not sit flat on the base of the Bissell, but kind of flops around, out-of-place, and the unit will not dispense water properly, though it may leak it.
If ithat piece is gone, it should be possible to reorder it. Start here:
If that's not the problem, if you have all the water tank pieces, I wonder if perhaps you unscrewed one black screw to remove the clear front plastic piece from the base of the unit-- the piece through which water is drawn up off the carpet-- to clean it, and didn't get it back into place properly. Make sure the two hooks at the bottom of this piece are inserted, one per side, into the clear plastic side pieces at the bottom. If the hooks are not secured properly, before trying to reinsert them, unscrew the black plastic screw so you don't break the tab. It's easy to miss getting these one of those hooks back in right, but it's important to do that before trying to screw the clear front plastic piece into place.