I had experience doing this as a kid. The wagon is too unstable, wheels are too close together and a sharp turn at anything considered speed will tip it over and if not carefully "hitched" somehow to prevent it - torque the bike over and possibly pitch you off. The front wheels are typically connected via a solid axle that pivots and makes it unstable in most ANY turn. Small solid rubber tires (unless you get a beefy garden cart type one with pneumatic tires) will also bounce and jar everything you carry in it.

I had a massive load of stuff in mine, tore off down the block went around a corner and it flipped and since I had it cinched up tight to the back bars of the rear banana seat stays it took me down with it. Still have the scar on my knee 40+ years later to prove it.

And as already pointed out they can oscillate if you reach just the right speed to create a harmonic and the thing will not stop doing it till your damn near stopped dead. It would be better to try a 2 wheel garden cart or a proper bike trailer, maybe a converted kiddie carrier trailer.

They're alright for pulling it if Wally and the Beaver are hauling groceries home from the corner grocery store at 5mph but anything higher may be a disaster.