Shampoos that irritate sensitive scalps usually contain sodium lauryl/laureth sulphate (SLS) or ammonium lauryl/laureth sulphate (ALS). A lot of people can't use SLS but ALS is okay, while some people can't use either.
I have super-fine hair and became allergic to SLS and ALS while I was pregnant
and took a year to find a decent replacement shampoo.
If you can use ALS, the Faith in Nature shampoos are very good. They clean well without being too harsh, have good eco credentials, and come in a range of flavours scents incl fragrance-free.
But if you can't use SLS or ALS, then some gentle baby and children shampoos leave them out and use things like Cocamidopropyl betain instead - they're don't clean as effectively as conventional shampoos, but rinse and repeat usually helps. However, better (IMO) is a soap-based shampoo bar - the best one I've found is from the Friendly Soap Company, and it cleans my hair really well, far better than the baby shampoos, and doesn't dry out my skin so much either.
None of the above shampoos contain silicones, so shouldn't weigh down your hair, but if weighing-down is a problem, be careful your conditioner doesn't contain silicones either.