Can you have a normal life with a herniated lumbar disc?

Yes . 75% are better nonop within 2 to 12 weeks from onset of leg pain. 90% get better without surgery. However , lifestyle changes are important to minimize recurrent problems: not smoking, regular exercise and weight control. If you need surgery, 90% do well, too especially if they follow the same advice.
Yes. Depends on your definition of normal life iin some countries back ache not considered a health problem(ae)volvo award winning paper early 90's many disc herniations resolve on their own or cause no limiting symptoms.