I applied tobradex (tobramycin and dexamethasone) on my eyes and my eyes burned and turned red. Is this normal?

No. If your eyes burned and turned red after putting the tobradex (tobramycin and dexamethasone) in, you should stop using it and tell your ophthalmologist. You may have an allergy to it. There are other eye drops that can be prescribed that are antibiotics and steroid combinations. It's important to follow up with your doctor so you can continue to have treatment.