The bushings are shot and my friend got if for free from tec school, it’s toasted already but I’m going to try to get an oil pump
I don't think you're quite understanding what I'm trying to get across.
The sump in that little banger is a splash lube case. The engine internals are splash lubricated, meaning there is no oil pressure and no pump. If you add a pump to feed the turbo, you're just going to draw down the level in the sump and you'll be looking at seized con rods and a window in the side of the block in short order. You'll also be adding heat to a sump that already can't really cool itself as it is.
You'll probably suggest a second or separate reservoir next, but that is just going to create it's own set of problems to overcome.
Dump the turbo and go back to messing with pulley diameters if you want speed (you're only going to get so much out of a lawn mower anyways). All the turbo is going to do is turn that little one lunger into aluminum scrap.
Find yourself an 820 transaxle. They're about the strongest lawn tractor transaxle you can get (I've got one on the shelves in the garage) and will bolt right in like your orginal (blown up) one. They're strong because they are rated for ground engaging attachments (IE: plow, harrows, etc), no other aluminum bodied transaxle is rated for that. The cast iron transaxles in the older tractors are stronger (IH, cub cadet, etc) but they usually have a different input that your vertical belt drive engine won't interface with, As a minimum, most use a horizontal shaft engine. If you want to go that way, you're better off just buying a complete tractor to build up. If you go that way, look at the websites that build the Kohlers for lawn tractor pulling. But you're still not going to get a lot of speed out of them. It's gearing as much as anything else. Garden equipment is built for low speed torque, not top speed.
Now, if you don't care if you destroy that little single cylinder modern engine, turbo away.....heck, at this point, might as well spray it with NAAAAWWWZZZZZ too.
But I'm guessing you've already mortally wounded it so anything you do to it now is just going to be wasted effort.
Sorry to sound negative and not trying to bring you down, I'm just being realistic about the design you're trying to overpower. Just keep in mind that speed is primarily a function of gearing, not HP. With enough HP you can overcome gearing, but we're talking ridiculous amounts of hp that you won't get from a lawn mower engine. Even then, you need to be able to rpm an engine to overcome gearing limitations and that little lawn mower engine redlines somewhere around 3600 rpm. You can sometimes press them to 4000 rpm, but much more than that and you're be firing con rods and pistons across the yard.....they're just not designed to make a lot of power.
A drivetrain is a system. you have to build all of it for the task you have in mind. From the water pump right back to the tires and everything in between. That's engine power characteristics, power transmission methods and final gearing (that's both axle gears and tire size). You can't just "improve" one part of that system and not expect it to negatively affect the rest of it....