Air carrier pilot reported a CFTT event while on a visual approach to DEN airport. The flight crew received a terrain warning and ATC also asked if they had the terrain in sight prompting them to manually fly the aircraft to a normal landing.

Narrative

We were assigned 170 kts to DYMON; flying level at 7;000 ft; configured gear down; flaps 30 and cleared for the visual approach. We backed up the visual approach by the ILS 35L. Upon reaching DYMON I retarded the thrust to idle; asked for check spoilers and flaps 45. I selected higher descent rate to intercept the glide slope. This caused terrain warning light but not aural warning. At the same time I switched to green needles without selecting heading mode first; which caused the airplane to enter a right turn. The captain suggest to fly manually which I immediately did. At 1;000 ft. our descent was stabilized. But gusty winds caused performance fluctuations; which I corrected for constantly. We made a normal landing.We were cleared direct DYMON with an assigned speed of 210 kts; expect the visual approach. We decided to decent to 7;000 ft. for DYMON; to be able to slow down for the final approach segment. Next we were assigned 190 kts. We leveled at 7;000 ft. approximately 8 miles for DYMON and were assigned 170 kts to DYMON. Approach asked 'do you have the terrain insight'; which made us realize that we descended to early to 7;000 ft.Cause: Loss of situational awareness caused the too early descend to 7;000 ft. The high approach speed and high work load at our selected descent point caused the automation error. Suggestions: A detailed briefing and planning of actions in areas of workload convergence could avoid a task saturation at the descent point. Increased situational awareness should be applied to recognize the built up of possible accident chain and to initiate a go around at the appropriate time.