If the parts had serrations (before they were worn off) that implies that the bite or interlock of the serrated faces were factored into the design, and you will not be able to tighten it enough now that they're worn smooth. You might be able to relate the the difference if you ever tried gripping with pliers whose jaws were worn.

You might try putting some coarse lapping compound (essentially grit mixed in grease) on the parts and clamping with as much force as you can short of stripping or snapping the bolts. (You might be able to beg some lapping compound from an auto mechanic, or engine rebuilder). The lapping compound increases the holding strength by biting into both parts creating a mechanical interloc between them (like spreading sand on ice for traction).

If you try this route be sure to position the bars exactly where you want them before tightening, and telling your son to tell you immediately if there's any slippage. This is critical because any relative movement causes the grit to make small grooves, and lose the ability to hold fast.

If the lapping compound doesn't work, your only other option is to replace both the handlebar and stem (the part from the bar to the fork).