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Breast milk - squirt in the eye area. Also cooled boiled water and make sure you don’t use the same cotton wool swab on both eyes.
Either colostrum/breast milk on the eye or a cooled down moist tea bag (sounds random but helped!)
If it is actual conjunctivitis rather than just a gooey eye, only drops will help. My baby had conjunctivitis at 3 days old and was prescribed drops from the GP. She ended up on them for 10 days as the infection took a little while to clear but then was all fine. Drops weren't an issue to administer whilst she was so little. But breast milk/sterile water will not sort the problem if it is a bacterial infection. If in doubt always worth getting it checked out as GP will be able to tell the difference
You can get antibiotic drops, maybe even straight from the pharmacy, but they won’t help with viral conjunctivitis, only bacterial and they look similar. Also babies hate them. We had conjunctivitis 3 times, used drops the first time (it was bad, eyes glued shut in the morning and swelling/redness) and little one screamed so much from the drops it was awful. From then on I cleaned his eye very carefully with cool boiled water, and it helped them clear within a few days. The skin around the eye is very delicate so if you notice redness you cab put a little baby Vaseline on