The development of turboprops and jets were going on more or less concurrently before and during the initial stages of WWII -their first test runs were within years of each other. You have to understand that there was a war going on- any (significantly) better aircraft engine would be the difference between life and death; and that would be the one that would get the resources. The turboprop was not to be that engine. During development, the turbojets showed improvements and promise that the turboprops never did.
Turboprops have propeller but the engine was completely different- instead of developing another type of engine (which will certainly be less powerful compared to the present ones) and making it to run the same propeller, it is easy to see why the concept of an engine without propeller is more appealing- for similar configuration, you have to show a significant improvement downright, while the new idea can survive (atleast initially) on promise. Add to this the fact that the piston engines reached their peak during WWII in performance and power, it would have been a tall order for the turboprops to better them.
Generally speaking, while turboprops of today may be faster than the piston engined aircraft, it was not necessarily true till the Soviets developed the Tu-95. In case of jet engines however, the difference was substantial. One of the initial problems in Me-262 was that it was too fast for prevailing tactics. During their introduction, the jets didn't offer any meaningful increase in flight ceiling.
For a turbprop, you need three parts- a turbojet, a gearbox and a propeller. A turbojet eliminates two of these components and makes the the engine simpler to construct (even compared to some piston engines), an important consideration in wartime.
At the time of the war, the primary thrust of engine development of improving the performance of combat aircraft- and the jet engines brought a paradigm shift in this regard by increasing speed, which the turboprop was never able to do. Even today, the turboprops are not popular for their power, but for their endurance, and their application in combat role is rather limited; in a situation like WWII, where the combat aircraft were paramount, their ascent would have been inconceivable. It was not tunnel vision, but facing the reality of appropriating the limited resources for surviving today and winning tomorrow.