Your turbo is a compressor that allows your engine to process more air.
By processing more air and by injecting more fuel you can increase power on each engine cycle.
So yes, you will have more power for the same engine at any levels.
Well... That's the physics of it, but the reality can be somewhat different.
In many light aircraft engines, the role of the turbo is much more to maintain power up to the maximum permitted for that engine, as altitude increases, through a much higher altitude. Generally little use is made of the available increase in power at sea level. This is because most turbo installations were accomplished as changes to existing engine designs, which were already operating at the limits they were designed for. Increasing the maximum power output of existing horizoontally opposed aircraft engines has not been universally successful.
Most turbo systems employ mechanical means to assure that the pilot does not extract more than the maximum permitted power (which would usually be very close to the non-turbo version of the engine at sea level). A few engines do not have such a system, and there is high pilot work load during takeoff to set the power carefully. An "overboost" mistake can be damaging and expensive.
Turbo aircraft are great, if the normal operation for the aircraft is above 5000'. Other than that, turbo systems are heavy and expensive, and of little benefit near sea level.