What is the most likely diagnosis for a 4-year-old child presenting with burning and itching in both eyes, a runny nose, red eyes, and swollen nasal turbinates?

The presentation of burning and itching in both eyes, a runny nose, red eyes, and swollen nasal turbinates strongly suggests viral conjunctivitis. This condition is commonly associated with upper respiratory tract infections, often caused by viruses such as adenovirus, which could explain the runny nose and swollen nasal turbinates. The symmetry of the symptoms in both eyes, along with pruritus (itching), further supports this diagnosis, as viral conjunctivitis typically presents bilaterally rather than unilaterally.

In contrast, herpes keratitis usually presents with more severe eye pain, decreased vision, and potentially a history of herpes simplex virus infection. Corneal ulcer would generally cause significant eye pain, photophobia, and possible vision changes, and often presents with an ocular history that is more acute and painful. Bacterial conjunctivitis typically involves a purulent discharge and may present unilaterally, especially in its early stages, which is not aligned with the symptoms observed in this child.

Given the combination of symptoms, particularly the ocular symptoms alongside nasal congestion and discharge, viral conjunctivitis is the most appropriate diagnosis.