Is it normal for my eyes to be bloodshot when wearing contact lenses?

No. Bloodshot eyes are a sign of irritation. You may be experiencing dry eyes, allergies to the contact lenses, contact lens intolerance or even infection. Stop wearing the lenses immediately, and see an eye doctor.
No. The redness is a symptom of underlying irritation - this may be caused my a myriad of reasons including dryness, fit, allergy, etc. Best to temporarily stop wearing the lenses and see an ophthalmologist.