There is a critical time period during a take off when the aircraft is at maximum risk. If an engine fails before rotation (i.e. before the wheels leave the ground) an alert crew can stand on the brakes and use thrust reversers. The aircraft may get dinged up, but there is a reasonable chance the crew (and passengers) will survive.
If there is an engine failure after rotation but before the aircraft has gained sufficient altitude, unless there's a big, flat field next to the runway a crash is almost inevitable.
When an aircraft turns, it will lose altitude unless the crew compensates by adding power. An aircraft without power and sufficient altitude cannot make the turns necessary to go all the way around to land on the runway they just left.