I'd say you should get him out. It wouldnt be so bad except they are really dirty (so watch for an ammonia spike) and they also need to get out of the water for at least 2 hours a day and bask in UVB. Your aquarium lights probably arent sufficient.
You could always do a little DIY project and make an above the water level basking area and raised canopy..could be cool
Looks like a common Musk turtle.
They need deep water, also need lots of hiding places and places to hold on. Your also going to need a basking area, with a UVB and UVA light.