Freud had an extraordinary influence on 20th century thinking, but it is a mistake to characterize him as having general acceptance within psychology. Freud was a psychiatrist/neurologist (the two fields had not separated at that time), which is a medical subspecialty. He was not a psychologist, which is a scientific field, and in fact had great difficulty understanding some of the basics of scientific methodology (there is a fantastic series of letters between Freud and Karl Popper where it is clear that Freud does not understand the basics of falsifiability and hypothesis testing).

Although Freud's theories piqued the popular imagination, psychotherapy was predominantly a psychiatric specialty for the first part of the 20th century, and was not a major part of scientific psychology. Scientific psychology was dominated by the structuralist, functionalist, and behavioral schools and clinical psychologists predominantly practiced assessment, not therapy. Psychologists would begin to practice psychotherapy in significant numbers because of WWII, which created a huge demand for psychotherapists in the U.S. military and VA medical systems. Initially, academic psychologists were retrained in clinical psychotherapy to meet the war effort and would practice under the supervision of psychiatrists, who predominantly practiced psychoanalytical techniques derived from Freud and his proteges (Jung, Adler, etc.). Clinical psychologists practicing psychotherapy during the post WWII years would develop new psychotherapy styles that were not part of the psychoanalytic tradition, notably the humanistic and behavioral traditions. This transition of psychotherapy fields greatly accelerated after the introduction of effective psychopharmaceuticals led to a widespread exodus of psychiatrists from psychotherapeutic practice in subsequent waves during the 1950s (due to antipsychotics) and 1980s (due to SSRIs). The change in balance between psychologists and psychiatrists practicing psychotherapy was also paralleled by decreasing emphasis on theoretical orientation and an increasing emphasis on psychotherapy process variables. By the time that clinical psychologists outnumbered psychiatrists in psychotherapy in the 1990s, the majority of practitioners would describe their orientation as "eclectic" and surveys would demonstrate that the actual techniques that most therapists would choose would have little relationship to the theoretical orientation that they claimed to adhere to. Although psychoanalysis is part of the clinical psychology practice tradition, it is the subject of great debate.

For further reading on this matter, I would suggest Leahey, Hergenhahn, or Schultz's histories of psychology.