Mingalababya,
Just remember, that whenever an aircraft is in a climb, lift will always be less than weight, reagrdless of whether the aircraft is turning or not.
When an aircraft is turning during a climb, the reduced amount of lift for the climb is being compensated by excess thrust, but of course more lift is required for the turn. This 'extra' amount of lift can be found, but the vertical component of that lift (for the turn) will not be able to equal the weight (pointing straight down towards the ground), whilst the aircraft is in a climb.
Hence the reason that climbing turns are always taught to be carried out with only a small amount of bank (around 15 degrees 'ish') depending on where in the world you are.
If anyone thinks different, please let me know, am always happy to be proved wrong and learn more.