Why were turbojets the first kind of turbo-engines, instead of turboprops?
Propeller planes already existed, and turboprops were an answer to a question that was not asked in times of war. The main driving factor behind development of the turbojet was to provide great climbing power and high airspeed for intercepting enemy bombers, which the turboprop would not have been able to do. Fuel economy and engine life were of secondary importance, the Junker-Jumo 109 of the Me-262 lasted only 10 flight hours!

The piston propeller engines were very sophisticated towards the end of WWII, the super-charged Merlin piston engine powering the Spitfire developed over 2000 hp.



But the Me-262 and the He-178 were way faster than the Mustangs &amp; Spitfires that escorted the bombers, and could only be shot down when lined up for landing. From the wiki: 


  Too fast to catch for the escorting Allied fighters, the Me 262s were almost impossible to head off. [Note 7] As a result, Me 262 pilots were relatively safe from the Allied fighters, as long as they did not allow themselves to get drawn into low-speed turning contests and saved their maneuvering for higher speeds. 


Turboprops would not have resulted in such an advantage, since the propeller is the limiting factor for airspeed. The lighter weight of a turbine would have been beneficial: the Merlin weighed 600 kg, and for instance the PW123 turboprop delivers 2,380 hp and weighs 450 kg - but only beneficial in the lower speed turning performance which the jet fighters learned to avoid.