L4-L5: compressed, 8mm one side- other 16mm. L5-S1: bulging disc with a 5mm mass effect. Shooting pain down my leg & numbness. Will I need surgery?

Differs. Each person is different. The overall response to conservative therapy (Physical therapy and Epidural injections and meds) depends on several factors: your weight, your core strength, your height, your genetic spinal canal width/height, etc. If you are short and fat, you are much more likely to need surgery with the described disc. If you are taller and slimmer, you are less likely.