Word for: "Repeatedly doing something you are scared of, in order to overcome that fear in time"

This technique of overcoming phobias, in psychology, is referred to as "desensitisation".

Note that the technique itself is "desensitisation" (a single word, as requested). The two-word "systematic desensitisation" specifically describes doing this desensitisation in a systematic way (e.g. starting small and increasing, rather than doing it arbitrarily or randomly), thus making it appropriate for use in clinical practice.

This wikipedia article gives an almost identical example, but with snakes:

A client may approach a therapist due to their great phobia of snakes. This is how the therapist would help the client using the three steps of systematic desensitization:
[Wikipedia]

One part of the CBT treatment process that's often used to treat simple phobias involves gradual exposure to your fear, so you feel less anxious about it. This is known as desensitisation or exposure therapy.
[NHS UK]

One possibility is habituate:

ha·bit·u·ate

verb

make or become accustomed or used to something.
"she had habituated the chimps to humans"

From the NIH

The term “habituation” in the field of stress neurobiology refers to the reduction in physiological responses elicited by an nth exposure to a repeated homotypic (same) stressor in comparison to the large responses elicited by acute exposure to that stressor.

Another is desensitize: From the NIH

de·sen·si·tize

verb

make less sensitive.
"creams to desensitize the skin at the site of the injection"
    make (someone) less likely to feel shock or distress at scenes of cruelty, violence, or suffering by overexposure to such images.

I suggest acclimatize. Lexico has

acclimatize (British acclimatise)
_VERB

1 Become accustomed to a new climate or new conditions; adjust.
Getting acclimatized to life in Glasgow was tough at first; even after all these years away he still gets lonely and homesick.