A few months ago I was given an injection to treat carpal tunnel syndrome. Since then I've had discoloration and change in skin texture, is it normal?

It's not uncommon. Carpal tunnel injections usually contain some steroid solution. Steroids can depigment and/or thin the skin. The subcutaneous fatty tissue can also become atrophied with steroid use. All of these affects are typically temporary, but may take up to a year to resolve.
Side effect. Corticosteroid injections can cause skin discoloration (hypopigmentation) and change in skin texture.
Side effect. Atrophy and pigment changes are a side effect of steroid injection, especially if the steroid is placed close to the skin. The effects typically will resolve with time, but may last over 1 year.