As u/Georgy_K_Zhukov noted, I wrote a previous answer that you might find useful that focused on why and how the Soviet Union gained and retained such a notable lead over the Western world in phage therapy. I'm going to give you three different related answers that explain why phage therapy was slowly rejected as a fringe technique for addressing bacterial infections in the West: we didn't really know what phages were, they were not consistently efficacious like antibiotics, and a few unfounded ideas got stuck in the literature.

For those following along, bacteriophages are the viruses that infect bacteria. Like human viruses do to our cells, they get their genetic material into bacterial cells, convert them into factories that make new bacteriophages, and then generally lyze the cells to release their progeny. With some bacteriophages making more than 3,000 new bacteriophage particles per cell, they grow rapidly at the expense of their bacterial hosts - killing up to one in five organisms that die every day.

We didn't really know what they were.

Bacteriophage were first described in the literature by Frederick Twort in 1915 . However, excitement with the possibilities of bacteriophage can be said to have begun six months before at the beginnings of the Great War with Félix d'Herelle, an infamously stubborn self-taught microbiologist at the Pasteur Institute. He was sent to Maisons-Laffitte, a Commune in France only 50 miles from the Western Front, to investigate an outbreak of dysentery among 10 French mounted infantrymen. Returning with samples, he described a soon eponymous novel 'bacillus' bacteria. However, in his investigations of this bacteria over the next 18 months, he found that some seemingly sterile Chamberland filtrates of it were capable of effecting the lysis of another dysentery bacillus (likely Shigella), destroying their cells. This demonstrated that whatever the cause of the lytic activity was, it was much smaller than bacteria, or even what now know to be the wavelengths of visible light. In one of the great scientific works of the twentieth century, translated here, D'Herelle described in two short pages the experiments that he performed showing that this lytic property could be serially passaged from one culture to the next by transferring 10^(-6) dilutions fifty times. Similarly, he showed that no dilution of these lysed cultures would produce hazy subinhibitory growth when plated over a lawn of bacteria, like any antibacterial toxin would, but instead would display a number of clear glassy plaques equal to the concentration that would lyse a liquid culture. From these observations, D'Herelle radically intuited that he had discovered un microbe invisible antagoniste des bacilles dysentériques," described it as un bactériophage obligatoire," suggested that his other bacteria would be found to similarly be infected by these pathogens of pathogens, and (perhaps too radically) posited that these bacteriophage were the true agent of natural immunity.

However, while d'Herelle was met with immediate excitement, his model for explaining the bacteriolytic action he observed was also met with rapid skepticism. Indeed, Tamezo Kabeshima of the Pasteur who quickly put this lytic action to work keeping rabbits alive long enough to produce antiserum against Shigella, soon published a paper suggesting that these “microbe filtrant bactériophage de d’Herelle" could not be living microbes as they were stable at 70°C and could be precipitated with acetone, alcohol, or ether. Thus, Kabeshima suggested that lytic property wasn't the result of a microbe at all, but instead much more closely resembled consequences of enzymatic activity. This hypothesis seeded a controversy that would erupt into a bitter feud between d'Herelle and two Belgian scientists, Jules Bordet and André Gratia, over the nature of bacteriophage and the priority that should be given to their discovery.

This may not have been helped by the aggressive way in which D'Herelle suggested that these two founders of the field that we now know of as immunology may not have discovered the basis of immunity (why people who get sick from a disease tend not to get sick from it again). While D'Herelle may not have been entirely wrong about bacteriophages playing some role, the questions that Bordet and Gratia brought up with D'Herelle's model were also entirely valid. Were bacteriophage living microbes, simply lytic enzymes, or a lytic principle of bacterial origin that could induce a "nutritive vitiation" that would produce more of the lytic principle? (See more description here) The question appeared to be partly solved by the resolution of the structure of bacteriophages by Ruska as well as Peankuch in 1940 through some of the first transmission electron micrographs ever produced. However, these questions about how we can describe the essential nature of bacteriophages did not have coherent answers until the molecular biology revolution, and honestly, they still echo to this day.

More can always be said, but /u/bblasdel wrote about baceriophages and Soviet medicine in this older answer, which you may find to be of interest here.