Car jerks hard when you shift to drive or reverse

Hello and welcome to 2CarPros. From the description it sounds like the torque converter has locked up. That means the engine tries to spin the gears in the transmission all the time with no slippage that would normally occur if it was operating properly. That would cause all the symptoms you have. If it was an electrical issue it would likely still start okay as the hydraulic pressure takes a second or two to build once the engine starts.
With 200,000 on it I would say it is due for either a rebuild of the transmission or replacement with a used one depending on the overall condition of the vehicle and if it has any other issues.