How long have you had him? It's not usual at all for a new turtle, especially a baby, to take awhile to get comfortable enough to bask. It can take weeks in a new habitat.
Have you measured the temps with a thermomter? Sometimes people are surprised when they get actual readings with a thermometer. Try checking the water temp and the temp on the basking platform with a thermometer if you haven't already.
Another common reason turtles don't bask is that it's too hard for them to climb up on the platform - ramp too steep, too slippery, etc.
Another reason they sometimes don't bask is that the light is hurting their eyes. What kind of light bulbs are you using and what's the distance form the basking platform?
On the soft shell, baby turtles normally have somewhat flexible (not floppy) shells. They gradually harden over their first year of life, with the last part to harden being the back near the tail. It's possible that what you're noticing is just this normal flex in a young turtle's shell.
