Without pictures I can tell you that you are doing fine. If grass comes up that easily it is not rooted well. Rake rake rake, remove debris to the compost pile. You will have to reseed. Does this little lawn get full sun or is it in the shade or is part of it in the shade most of the day? Do you get puddles when you water or it rains?
Why did you think you needed to thatch? Normally if you can see a matrix of roots an inch thick or deep then thatching is recommended. This gets rid of this matrix that is stopping proper water infiltration as well as added chemicals necessary for the grass plants. Thatching IS brutal and kills lots of grass.
I'd mow THIS TIME ONLY very very short after you've raked as much debris as possible. (Most mowers act like a vacuum cleaner). Reseed with a hand held 'spreader' over the entire lawn. Make sure the seed you buy is ZERO weed seed. Try to find seed for shade as well as sun. After seeding, rake again lightly. I'd get a roller (rent this and fill with water for weight) if your lawn bed is pretty roughed up and roll the whole mess until there are no depressions or bumps. That means to add more soil to the depressions and rake the bumps level, then roll. If necessary do this again until you have a level bed. Hopefully this lawn already has a bit of a slope so that it drains somewhere...you can assist this drainage of 1-2% at this point. Do not worry about the live grasses. Thatching is kinda sorta starting all over again to repair the bed of your lawn so don't sweat killing live grasses. Remember thatching is NOT MAINTENANCE. It is the second to the last trick to renovate a lawn. The next is a sod cutter and completely removing any grass and roots.
Growing from seed can be a bit of a headache. There are millions of weed seeds in the soil. No matter what you do this is a fact. The trick is to prevent having to ever thatch again and maintain a nice grass mix without any weeds. Following are the basic must do rules:
Mow no shorter than 3" (cool season grasses)! Critical. Not 2 1/2" Not 2"...not 1 1/2"!! This length helps to slow evaporation of water and shade the lawn bed. Shading the soil with 3" grass will prevent most of germination of weed seeds. More importantly our cool season grass species have genetically LARGE ROOT SYSTEMS. If the top growth is NOT 3" then the plant doesn't make enough FOOD for the roots. Fertilizer is NOT FOOD. Plants make their own food and if there isn't enough photosynthetic top growth the roots aren't being fed correctly which makes for a whimp, unable to compete grass. Not vigorous, not healthy, not deep green but a whimpy crop of grass. The following is how to change this...
Mow a minimum of once per week with SHARP mower blades. Remove the clippings to the compost bin. I've yet to find a mulching mower that can cut blades of living grass small enough to not cause thatch eventually. Grass grows slower the longer it is. So each week you'll be just buzzing the top for a level field. Far less clippings and with such a smallish lawn I'd bag the clippings and take to the compost pile anyhoo.
Watering is a critical component for training your lawn to be drought tolerant as well as eliminating an awful lot of weeds that are shallow rooted. When you water you water DEEPLY. 4-6" deep within the profile of your lawn bed. Use a shovel. Then the trick is to let that lawn DRY OUT until (this is the best way I've found to know when to water again) when you walk on your lawn your footprints stay. The grasses you bent down stay down. THEN and only then should you water again and very deeply. Work up to 1" per week of water (use straight sided cans to measure how much water you are putting on the lawn bed...all over the lawn to see the uniformity of watering as well) Depending on your soil type this could take 20,30 minutes or an hour to get the soil wet at least 4 -6" deep. As you allow the lawn bed to dry out, the grass roots are growing deeper to get at the moisture. By doing this you are allowing your cool season grasses to develop the deep and large root systems that are genetic. (And have 3" top growth at all times). Between watering sessions your grass will be lush and green because of the deep roots. The weeds if they germinate at all (usually very shallow rooted) will die of water depravation. This includes moss...and btw moss is not a competing 'weed'...it is an opportunist and if there is enough shade and moisture and there is bare soil because the grass can't stand shade and whimps out...you'll get moss. Better than bare soil...thank goodness, huh?
Then there is pH. If you have lots of rain or water an awful lot or if you've used peat moss...the lawn bed soil will probably have a low pH meaning too acidic for lawn grasses. Get a soil test. Lawns thrive within 6.5 and 7.0 pH. If it is below 6.5 I would then add lime. I would do this in stages, you do not want to put too much lime on at a time. Get a soil test, best by your cooperative extension agents (or a University program for gardening/composting/preserving food). There are 'kits' to do these tests but kinda iffy. Do not add lime without testing first!
Once per year or twice even, aerate your lawn bed. This means renting a core aerator that takes soil plugs out of the bed. Leave them sitting right there on the lawn to disintegrate. Do NOT remove the plugs from the surface of your lawn. Do not use tine type aerators as they actually compress your lawn bed soils.
If you get a mole or two (usually no more than a little couple) they will aerate, control soil insects like crane fly grubs as well as 'top dress' your lawn with those little piles. Just rake them down. Those little guys are doing you big favors for free.
OK and here is a biggie. Fertilize with extended release organic fertilizer. Do not use 'fast' stuff by Scott's or Ortho...I was blown away with using Dr. Earth's lawn fertilizer. A bit more expensive than the fast commercial lawn fertilizers but worth every penny. You can easily cut fertilizing down to 3/4 so the cost is mitigated. This particular brand also includes bacteria that are responsible for breaking down thatch! There are others and when you try a great fertilizer that is organic (I am very careful with 'organic' or 'natural' but this really makes a difference in lawn fertilizing) you will flat out be blown away. Make sure the formulation is appropriate for the time of year, ie) higher nitrogen in spring and summer and lower nitrogen for the fall.
Mowing height, sharp blades, regular obsessive mowing or buzzing the tops of grasses, your watering technique and root training, aeration once per year and great fertilizer...you should never have to thatch again, nor get weeds. Make clean edges and you'll have the healthiest, prettiest little lawn in your neighborhood!