If a patient gets l2 vertebral fracture will it be treated with bed rest?

Depends: on how much pain is caused by being upright or moving around. Generally movement is better than no movement if it is tolerated.

Vertebral fracture: Depends on how bad the fracture and is there is any spinal cord compression or not. Usually , the compression up to 25% can be treated conservatively provided no neurological symptoms . MRI and CT spine will be required to assess the extent of injury before making a rescission on conservative management .

Vertebrae fracture : Management depends on the kind of fracture and if there are any neurological complications or not .

As per dr singh : If the injury is stable then usually they try to get people moving. If it's unstable then initially bed rest/immobility will be required until stabilised with treatment. Ask your doctor directly for advice and as per dr singh it may well need extra imaging to sort it out