How old are the broccoli seedlings and how tall and wide are they?

I don't necessarily think they need to be fed. I normally don't feed mine while they are still in flats.

My best guess is that they either have become etiolated because of low light levels (too many cloudy days at my house, so I'm guessing the same might be true at yours as well) in the greenhouse, or that the broccoli seedlings are experiencing temperatures that are too hot for them. Keep in mind that broccoli plants are cool-season plants and most cool-season plants, as seedlings, prefer to be grown at temperatures of roughly 50-55 at night and 60-65 during the daytime.

I don't know what your greenhouse is like, but yesterday, with it being 89 degrees outdoors and, despite having open doors and vents and 50% shade cloth on my greenhouse, the high in the greenhouse hit 100 degrees---shortly after noon yesterday. Even with a fan running, I couldn't get it down any cooler than the 80s. If I'd had cool-season plants in there, they'd be toast. My one tub of winter lettuce that is still in there was not amused by yesterday's heat.

All my cool-season plants are in the ground already, except for a few cool-season flowers that I haven't put into the ground yet, but I'm leaving them in the garden at the shady northwest corner so they stay cooler than the plants in the greenhouse are staying.

When you plant the broccoli seedlings, just plant them more deeply so that the etiolated stems are below ground. I'd try to harden them off and get them in the ground as soon as possible, even if they still are pretty small. It is getting really late in the cool-season to have cool-season plants that haven't made it into the ground yet. For every day that your broccoli seedlings remain in the greenhouse, they are missing out on prime growing time.

I started my cole crops very late this year and put them in the ground almost as soon as they sprouted. They were ridiculously small. All of the cabbage and broccoli plants are growing like crazy in the weather we're having now and actually look about the same as they always look at the beginning of April. We have been awfully warm, so they've grown so fast that they've made up for being started late. Now I am just hoping all the early heat doesn't make them bolt before they can produce a good harvest.

These are piracicaba and purple sprouting. I promised them to the new neighbor and I'm kind of embarrased to give them to him. I winter sowed some of these and those are in the ground, though not very big. I am going to quit trying broccoli in spring! You are probably right about the green house being too hot. I have old sheets draped across the top, so now mid day sun doesn't come through. I pulled most overwintered herb plants out yesterday. The greens I planted in Jan came out last week (and I had a salad yesterday) The green house will be the first stop for 2 flats still in the house. And I will probably move the tomatoes back in there for Friday night. I will be storing the plastic cover soon, though, as it will be too hot.