Assuming your lens is the Canon EF-S 18-200mm f/3.5-5.6 IS, you should never turn the focus ring manually when the AF/MF switch on the side of the lens is in the 'AF' position. As you have painfully learned, that can damage the focus mechanism. It's not just an adjustment that can be made, there are broken parts inside the lens that need to be replaced.

The EF 18-200mm f/3.5-5.6 IS was introduced in 2008. Like most non-USM lenses introduced before STM focus technology took over, there is a direct mechanical link between the focusing ring and the AF motor when the lens' switch is set to 'AF'. Forcing the focusing ring to move when the gearing is engaged with the AF motor puts more stress on the linkage than it is designed to handle. Canon includes warnings to not do this in the Instruction Manual for every lens that does not have 'full-time manual focus', which are almost exclusively USM and STM lenses. It's on page 4 of the EF-S 18-200mm f/3.5-5.6 IS Instruction booklet

The EF 18-200mm f/3.5-5.6 IS was recently discontinued in 2021, so Canon will still be providing repair support for this lens for a few more years (typically seven, but sometimes a little less if their reserve of parts run out). Canon repairs these days, at least in the U.S., are on flat rate tiers that include whatever needed parts. What is covered on each tier and how much each tier costs varies from one lens model to the next. So I'd call Canon customer service, tell them that you broke it by turning the focus ring when the switch was set to 'AF', and see what the cost would be for (probably) a Tier 1 repair (or whatever tier they say you'll likely need). Then you have to decide if that lens is worth spending that amount to restore to working condition.

If you're candid about what led to the problem, they'll have a pretty good idea what is broken inside the lens and how much they'll charge you to fix it, though they will not guarantee that estimate until they've actually examined the lens. They'll then send you an e-mail requesting you to approve or decline the repair. I've never had them change the estimate the few times I've had to send a lens to Canon Factory Service in the U.S. since they've went to the tiered system.