Assume the left engine failed and the right engine is at full power. The pilot has pulled his nose up and turned towards left. Is it possible for him to land the plane on a runway to his left? The plane is an ATR-72 turboprop design
No, no, by all means don't turn left. See also Why should you not turn in the direction of an inoperative engine?.
Transport category aircraft are capable of climbing out on one engine. It is a certification requirement for them. So if the left engine failed after rotation, the first thing the pilot has to concentrate on is to keep the plane flying and under control. That means ease the pull on the stick a bit to avoid stalling as the plane is not able to climb as fast and push the right rudder pedal to keep the plane flying straight¹.
Once the plane is in straight climb at $V_2$ speed, it's time to deal with the engine. First both pilots cross-check the instruments and the rudder position to determine it was the left engine that failed. Then they feather the propeller (auto-feather might have done it already) and cut fuel to the failed engine. If the engine fire warning is on and did not extinguish when fuel was cut, they discharge fire extinguisher.
These things are done according to standard emergency procedure that the pilots have to remember and train in simulator. I don't know the specifics for ATR-72, but in all aircraft it is roughly like above.
Meanwhile the aircraft climbs to safe altitude and then they should level off and let it accelerate a bit to get some safety margin for manoeuvring.
The plane is now in relatively safe condition. Even if engine was on fire, it is unlikely to spread while the aircraft is flying. So the pilots now have time to declare emergency and ask the controller for what options they have for landing. Whether they do a complete circuit and land back on the same runway or turn around and land in opposite direction or land on some other runway then depends on wind and where they are relative to the runways at the time.
There were many accidents where plane crashed after one engine failure during take-off because pilots started manoeuvring before they attained sufficient altitude and safe speed. It is better not to hurry, engine failure does not require it.
¹ Of course provided there are no obstacles in the way. For example if you were taking off from VQPR runway 33 and your right engine quit, you'd still have to do a right turn rather early to avoid the Mr. Smith's house right in the axis of the runway and quite a bit above it. But for any airport that require such avoidance (and there is quite a few of them in various mountains), the pilots are specifically briefed and trained for that airport which includes instructions on how to best handle engine failure during take-off and during approach and viability of the procedure is verified in simulator and with a test flight. The above answer just outlines the generic procedure.
The aircraft is able to fly on one engine. Sudden manoeuvring immediately after an engine failure on take-off is not advisable; the margin from the minimum control speed with one engine out (Vmc) is low and assumes a 5 degree bank into the working engine. The abnormal operating procedures of the ATR-72 advise the crew that:
The conduct of any emergency procedures or the analysis of any technical problems should not normally take place until:
a. The vertical and lateral flight path is under positive control;
b. Possible ground contact is no longer a threat; and
c. Altitude is above 400 FT (AGL)
When the above conditions are met the engine shutdown procedure should be followed. These actions are to be memorized by the crew instead of being read from a check list as they are critical.
After the engine is shutdown the crew can assess the situation and decide on further actions. Returning to the departure airport will be the normal course of action but if circumstance dictate so the pilot in command may decide to land on an alternative airport.
Following an engine shutdown a two engine aircraft should normally proceed to the nearest suitable airport, although there is no objection to the aircraft proceeding to the destination provided the increased flying time is not significant.
The quotes are from an ATR-72 SOP book I found on scribd
The PIC has the authority to do whatever he/she deems necessary to control the situation. Landing on another runway on the same airport is one of several choices. Not all airports have multiple runways, and if they do, not all are suitable for an immediate landing, but if the runway layout allows for it, then yes, landing on another runway is a viable option. But only one of many. I always impart in my ME students to consider landing on another runway in case of an engine failure upon take-off, IF (and only if) the runway layout is suitable for that. On most airports, landing on another runway is not an option.