Court TV broadcasts over the air and is free to watch on Roku, Amazon Fire TV, Apple TV and the Court TV app on Android. Plus, you can watch it live for free on Court TV's website. Trial testimony from previous days is also available on demand on the Court TV app and website.
You can search for the Court TV app in LG TV app content store to see if it's available. If not, LG TV does not support jailbreaking. Therefore, you may consider mirroring Court TV to your LG TV.