General aviation pilot reported a Class B airspace incursion when following a noise abatement takeoff in visual conditions from a non-towered airport. The aircraft received a terrain warning while turning and also entered the Class B airspace; then descended.

Narrative

While holding short of runway 11 I read a sign with noise abatement instructions that said to fly straight out 2 miles at best rate of climb before turning left to follow railroad tracks on downwind. I was planning to depart southbound. After takeoff I climbed at a high ROC but below my best rate of climb which would be around 4000 fpm in Aircraft X. I arrived at the 2 miles mark and proceeded to make a 270 degree turn to a southbound heading when I received a terrain/obstacle warning with an audible 'PULL UP! PULL UP! OBSTACLE!' alert in the middle of the turn. I pulled back slightly on the stick and increased my bank angle sharply to tighten up the 270 degree turn. I did not see any obstacle hazards in my path and when I looked at the altimeter I briefly saw an indicated altitude of 2000 feet and corrected immediately. The floor of the Chicago Class B airspace is 1900 feet in that area.I was unfamiliar with the noise abatement procedures at 06C and in the future if I find myself in a similar situation in a high performance airplane like Aircraft X at an airport with unfamiliar noise abatement procedures I will prioritize safety and airspace considerations ahead of noise abatement.