Furthermore to @Zeiss' answer, whenever an aircraft is steady-state banked, the lift will be greater than its weight. However, its speed will be constant; instead, the acceleration is centripetal and results in a circular turn.
Edit, clarification on pull up maneuver:
When an aircraft is pitched up via pitch control, and after the short-period mode settles (a few seconds at most), it will gain a lift imbalance greater than its weight due to higher angle of attack (AOA). Similar to a banked turn, since the net force is perpendicular to the horizontal velocity, it will be centripetal and results in a vertical circular motion. This is the early stage of the maneuver.
Since a higher AOA has higher drag, the airspeed will decay toward a new and lower trim speed. At the same time, as airspeed decreases, the net lift also decreases, so there is less force imbalance. As the decreasing airspeed undershoots the new trim speed, the aircraft will pitch down again. This cyclical exchange of energy is called phugoid and usually persists for a while (dozens of minutes to an hour) if left unchecked.
Eventually, however, the phugoid dies down (phugoid is usually stable in non-transonic regime) and the airplane is flying at the new trim speed, with lift equal to weight once again. If the airplane is in the front of the power curve, it will climb at steady-state; otherwise, it will descend.