Wider bars won't solve the issue. Have you determined if there is any side to side play in the wheels that might indicate loose or missing bearing in the hub? Possibly the headset is loose too. Very unlikely, but the spokes might be too loose.

I didn't do such a great job on my first rides both when I started with training wheels and later when the training wheels were removed. Crashes, scraped elbows and kneses were the norm.

In my case the bike was in good shape. It was just the kid, me, on the bike not doing the right things at the right moment.