Hi Al, welcome to the forum. Let's start with some common stuff. If I'm in your shoes, here is what I'm doing...
Inspect: What shape is the air filter in? I'll make adjustments with cleaning and oiling if I see fit. I'm going to check the airbox in case any critters made a home in there that will obstruct air flow. I continue forward by looking at airbox to carb boot then the carb to engine manifold looking for cracks or holes and fit and how everything is sealing, as well as the general condition of the rubber. As with tires, these parts are recommended to be replaced every decade or so because of rubber breakdown and deterioration. Pulling the seat and the side covers, I'm looking at the condition, routing, sheathing, connections of the wiring and spark plug wire and boot. While I'm in there, I'm changing the 20 amp fuse and the spare, because an oxidation layer can cause some funky stuff, and it's just too cheap and easy to do not to do it. Tighten the battery connections.
Easy, stupid stuff: Change the oil. Change the spark plug. Even if it's good. $5 for a new one, just change it. Eliminate it as a source of any problems. Then, even if I think the gas is fresh, I'm changing it for brand new stuff straight from the pump. I pour the old stuff into a gas can and then, if it still looks and smells good, it goes in my truck. Since the carb is probably suspect, when I put new, fresh gas in an empty tank, I'm going to add a bottle of Sea Foam, since it can work wonders, and might just work without a total carb breakdown and cleaning. The secret with an additive like Sea Foam is to run the bike and get it in and through the system, then let it sit, preferably overnight. Repeat. Repeat. With this though, don't run the bowl dry, leave it wet. Run the bike, shut it off with the key, turn the petcock off, let it sit. Do it again later. Again. Let it sit to work to breakdown the gunk, then run it through. Let it sit again.
Worth noting: The stock TW carb is set very lean from factory to get that EPA and MPG rating, but most change that to make it start and run better. The pilot circuit screw is underneath towards the front of the bike. The stock setting is .95 turns out from a light seat, but most find going to around 2.5 turns out is the sweet spot. If it's never been touched, then it is hidden under a factory cover which can be removed by making a small hole, then turning in a small sheet metal screw to bite the cover, then pulling on the screw with pliers or vise-grips.
Finally: A good carb cleaning with a full breakdown and cleaning of the body in a sonic bath with a solution of simple green. Inspect the internal parts and replace as necessary.
I'm trying hard to think of why it would only start when laid over on its right side. Here are my quick possibles and what to check... 1) The petcock is on the left, so once you open the petcock and get fuel into the bowl, something with the float and valve might not be quite right and flood. When laid all the way over to the right, the fuel may not get down the petcock tube? Turn on petcock for 15 seconds to get fuel into the bowl then shut it off. Keeping the bike upright, see if it will start. 2) Old, thick oil and a weak battery may not be letting the engine crank quickly enough to start and leaning it over put all of the oil over, reducing the load in the bottom end enough to allow the engine to crank faster to start? Charge/change/jump battery and change oil. 3) There is a nest or something in the airbox that obstructs the intake tube, but leaning over moves the object, allowing airflow for the bike to start? Check airbox. 4) A wire or connection is grounding out or not connecting, but when the bike is laid over it hangs free or connects properly? Check wiring and connections.
You'll want to check out our Stickies in the Technical Write Ups section... Technical Write-Ups
You may want the Service Manual... https://tw200forum.com/forum/technica...r-manuals.html
Before tearing apart the carb, get familiar with it here... https://tw200forum.com/forum/technica...ification.html
If you do open it up, you may need this... https://tw200forum.com/forum/technica...tor-float.html
Good luck. Let us know what you find.