My garden was not always organic in my early years, but has been so for the last 30 years or so. My main reason for going organic is knowing what is not in my vegetables.
Nearly half of my garden is various legumes, which helps to preserve fertility. I turn over the 20-25 bales of hay that I use for mulch each year, plus whatever leaves & lawn clippings I can gather. The neighbors & DD are helpful in that regard, giving me bags of shredded leaves in the Fall. All garden waste (other than infected waste) is also returned to the garden. such as bean hulls. Adding all that organic matter has been effective, both the gardens on my property & my rural garden (on a friend's property) have been gardened for about 15 years, and have shown only a modest reduction in fertility. I'm presently looking for an alternate garden site, so that I can place the present gardens in green manures for a couple years (and maybe find a better drained site for my garlic).
I miss the garden I had when I lived in San Diego. It was across the highway from a horse stable, I could take all the composted horse manure I could haul for free. I turned under 4-5" over the whole garden each year. It began as hard-baked desert clay, and was loose fertile loam when I turned it over to another gardener.
The biggest hurdle with organic gardening is the control of pests & diseases. Finding & growing resistant varieties is part of the solution. Encouraging insect predators, while avoiding conditions that promote harmful insects, is also important. For the most part, if I am patient, a healthy population of beneficial insects solves problems with aphids and caterpillars; lace wings, lady bugs, syrphid flies, and wasps are all present in large numbers. If I can get them up in time, row covers protect squash from SVB, and traps help to keep Japanese beetles down to controllable numbers. On the rare occasion that a particular insect or disease threatens to wipe out a crop, there are usually organic solutions... but push comes to shove, I would rather lose a crop - or not grow it at all - than resort to poisonous chemicals. The only exception I would make would be to save a rare variety whose seeds could not be replaced.
IMO organic gardening takes patience, persistence, and a fair amount of learning. It takes time to build up poor soil, and to build up a good beneficial insect population. If you save your own seed, successive generations may become better adapted to your conditions, and their yield will improve. I've witnessed this in several of the vegetables I save seed for; beans become more vigorous, okra has become increasingly resistant to the wilt which once ravaged it, and the DTM of some of my limas has been reduced by as much as 3 weeks.