Stronger frames yes but the root of the wobble is those rear isolator bushings/motor mounts or "donuts" as they are called. The azze end of the bike is literally hanging on 1/4" of soft rubber per side which allows the entire frame to articulate like one of those folding buses.

There are bolt on products that will undoubtedly help but the best solution IMO is the Glide Pro kit that replaces the OEM donuts and swingarm shaft and front motor mount. Put the kit on my '91 Glide and it's like it's nailed to the road now. When I wiggle the bars at speed all that moves is the front wheel instead of the whole bike flexing.

Jake is the owner/inventor and has sold thousands of these kits. Check out his informative vids on his website and educate yourself on the best solution.

 

Bagger Brace on my 08 FLHRSE4 incredible workmanship, and easy to install as long as the dog bone brace under the frame isn't bent. The bike is glued to the road.

 

You forgot to add "on a bike that he has no experience on and doesn't yet have the necessary skills to ride."

 

That is not entirely true! There is only scope for a very small lateral movement where those rubber bushings are. Bendy buses they ain't! Better they could be.

 

For me it was the tires... 09 Road Glide. Long fast sweepers 60 mph ++ A butt clench-er to say the least and the wobble always seemed worse on the right turns vs the left

Since the only way to prove it, it to swap 'em out which is very expensive on a gamble, most folks save this for last... I'll bet it's the #1 cause...

 

Had the same problem OP described. After a lot of time and money checking out many of the suspect problems other posters have mentioned, all it ended up being was the front wheel out of balance. Quick fix, never had the problem since. If the problem ever happens again, first thing I'm going to do is check the tire balance.

 

Tire pressure!

36 front

40 rear

Your working pressure may be a little more or less. BUT it does make a huge difference, especially cornering!

 

For your ride, all other things being equal, swing arm bushings good, front motor mount, tires, bearings, etc. a Bagger Brace or True-Track should fix you up. Good luck.

 

It's a byproduct of years of refinement, small "fixes" to an antiquated design. Plain and simple.
Ride down a straight road at a decent clip, and wiggle your bars. What happens?
First off, you'll notice a huge discrepancy between where your handlebars are pointed and where the front tire is going (not to be confused with countersteering). Next you'll notice as the handlebars catch up to the front wheel, a fairly stable wagg of the bike, traveling rearward. As the bars travel the other way, the same thing occurs, opposite. You can do this all day, at pretty severe input, and still not upset the bike to the point of creating a full on tank slapper.
This is inherent in all motorcycle design, mostly because of the gyroscopic effect, but also because of the forces induced by countersteering.
There are however a "stack up" of numerous other things going on in a touring Harley.
For starters, the designed in flexibility of the front forks, starting at the weak triangulation of the triple trees, traveling down small dia. tubes, into sloppy fork bushings with poor overlap, and terminating into a less then stout front axle. Luckily the front fender IS a stressed member and is essentially used as a fork brace (Remember that when modding the fenders guys).
Add the rubber mounted handlebars, and that is enough to get this whole ball of wax rolling.
The next part of the equation, is the frame. A center backbone frame is whoefully inadequate to stop any type of twisting from being translated front to back, especially when the engine is not utilized as a stressed member, and is also a big contributor to the wagg.
Getting into the engine/swingarm design, I'm referring to them as one, because even though they act in a vertical plain separate from one another, they act as one in the horizontal plane.
This is where it gets interesting, and this is where I think most derive a diverging conclusion. The engine/swingarm is NOT solidly mounted to the frame. It is designed to rotate in very controlled vertical plane, relative to the engines gyroscopic forces, by using a series of rubber bushings, and pendulum links to keep it in it's intended path, and to allow for the Harley rumble. The engine/swingarm and frame junction is also rubber mounted to allow a small amount of axial flex about it. This is because the pendulum links require a mall amount to keep them from binding, and also to isolate some of the engine loads from transferring too much engine vibration into the frame and rider.
Since we're here, the primary purpose of the bagger links, is to limit this axial rotation at the engine/swingarm/frame junction. Reducing the amount of flex that is translated into the dreaded wobble.
Now add to that, tire squirm, underdamped suspension, high CG, varying loads, wind updraft caused by the fairings, and countless other minor inputs, and it's surprising the things don't spit us off at every turn.
A few more things I'd like to point out. One, with all the various solutions described just in this thread, it's pretty obvious to me that one thing slightly out of harmony, bearings, tire wear, loading, bad shocks, etc. can really upset the apple cart.
Two, the methodical refinement of the whole system, has brought, what in terms of fresh design would be a nightmare to a designer, has in fact created a truly unique and fairly capable machine.
Flame on!!!

 

BTW, the newer 2009 - up frames addressed the axial flex by adding two rubber engine mounts up front, triangulating the engine in the horizonal plane.
The rest of the redesign was to reduce manufacturing cost. For the most part.