breastfeeding troubles

My daughter is now over a month old and for the past month we have been struggling with me having a really low supply and having to supplement with formula sometimes. I did everything everyone told me. I drank LOTS of water, I ate oatmeal for breakfast every day, I took all the meds and teas for milk production. I would pump after feedings between feedings and during the night. But nothing was really helping except when I started pumping my supply went SLIGHTLY up. Everyone on here just told me to put the baby on the breast and my body would know what to do. Well, that made me feel like an even bigger failure that my body didn't know what to do. We had our one month check up two days ago and apparently my daughter has both a tongue AND lip ties. I was told it was probably a big reason for my low supply (and I mean low when I wouldn't feed for 4 hours for her naps I would pump and get half an ounce at most. Yes I know a pump isn't as good at removing milk) so hopefully fixing this issue will help my supply go up. Every time I give her formula my heart breaks. I don't bash formula feeding mothers, but breastfeeding is something I always wanted in my parenting.