Losing one engine on takeoff can be worse than losing both. If you are flying below minimum controllable airspeed, critical engine out, that is worse. Many aircraft have a best rate of climb speed very close to the above. If you lose the critical engine and have found yourself at too slow of an airspeed, you will roll right over on your back. There is no recovery from it unless you have a few thousand feet below you to play with.
That makes sense..
Required Engine power is at absolute maximum during takeoff. If one engine fails in a 2 engine aircraft, and aircraft has not reached high altitude, it must be impossible to control the aircraft. It will turn sideways as this one did, with the good engine above the bad one.
My guess is the pilot turned off the good engine with a hope of a crash landing in the river. But just did not have the altitude to pull it off.