Biggest issue is probably heat. If you want cooked whites and runny yolk, without the crispy edges, you may need to turn down the juice. A heavy pan - cast iron - can also help even out hot spots. @SleepyWill has some good suggestions, both of which serve to cook the egg from both top and bottom. (When you’re cooking a slab of meat, you can get less overcooking on the outside with frequent flipping - thanks, principles of heat transfer! - but with an egg, that’s a recipe for burst yolk.) You can also cook the egg covered, heat on, as long as you check it frequently. A little moisture in the pan - a fresh pat of butter is ~15% water - should help.
My wife prefers softer whites, while I like 'em crispy, so I always cook hers first, then mine, as the skillet heats up. If you like yours fried in bacon grease, it makes for the best crispy edges when done in the bacon pan, or spoon the hot fat into another pan to get the bacony flavor without having to let it all cool down first.
Or make poached eggs. If you get lacy brown edges on a poached egg, you’re doing something spectacularly wrong. Or learn to make shirred eggs. Mmmm.
You can also reduce the amount of lacy edge by straining off the thin white through a fine mesh strainer before cooking. (Super-helpful for avoiding raggedy edges when poaching, too.) The older your eggs, the more the white thins, giving you that wide-spreading effect, and that’s always going to cook much faster than the perkier thick white.
Fun side note: @casuistry mentions the room-temperature v. refrigerator egg storage difference. I kinda doubt the temperature shift has much of an effect. But the reason UK eggs live on the counter and US eggs in the fridge is that US egg regulations require eggs to be washed before sale, which (at least for commercial eggs) removes the protective ‘egg bloom’ cuticle that surrounds the shell. Without that bloom, at room temperature, there’s a higher risk of bacterial infection, as well as greater moisture loss from the egg’s interior. So into the cold they go.