だいぶ良くなってきましたって英語でなんて言うの？

If you say, "I'm over the worst of it," it means that the worst part of your illness has passed and you are obviously 'on the road to recovery.' Similarly, "I'm on the mend now," or, "I'm feeling much better now," both indicate that your illness is on the wane and you may look forward to a swift recovery in the near future.

The verb recovering refers to the body's natural process of healing itself. One could say for example: I am recovering from my illness. This means that you are healing or that your health condition is improving.

Alternatively, the expression I'm feeling a lot better can be used. This is a common expression to indicate that your symptoms have alleviated.

If someone asks how you are feeling you can respond with 'I feel a lot better now" which means before you were not feeling good but have improved. You can also say "I'm feeling much better" which means you have overcome your illness.

Also,
I'm returning to my usual self again. (After the major heart surgery he had, he is returning to his usual self again!)
Recuperating = recovering from illness or exhaustion (She is recuperating after her recent heart surgery)
Convalesce = regaining health and strength over a period of time after an illness or medical treatment (He spent a couple of years convalescing from his skull fracture).
Simple words that can be used: regaining strength, getting better, getting stronger, getting back on one's feet, feeling oneself again, getting back to normal, returning to great health, on the mend again, on the road to recovery, healing, making good progress, turning the corner, making a turn for the better, bouncing back.

In the first statement, you have said that you have healed much better. You have used the verb 'to heal' in its past participle tense ' have healed'. The verb to 'heal' has several meanings but in this context, it means 'cause (a wound, injury, or person) to become sound or healthy again'.

In the second statement, you have used the verb to 'heal' in its present participle tense 'healing' because the healing process has not yet finished but is continuing to do so.

"(1) I'm feeling much better. ." You can say this if you are asked how you are feeling in relation to your illness.

"(2) My health has improved a lot.."
You can use this response if someone asks about the state of your health after an illness.

"(3) I feel a lot better, thanks to the medicine. "
Especially after taking medication that helped you to recover from your illness.

I'm on the mend - this means that you are starting to feel much better
and that you are over the worst of your symptoms.
I'm doing much better, thanks - this means that your health is improving
and you are not feeling as sick as you did before.

My health is improving' and 'I am feeling a lot better' are usually used if we have been sick or had an illness.

'It is beginning to heal' is usually used if we are recovering from an injury. (Starting to get better)

Q: How is your knee after the operation?
A: It is beginning to heal.

All of these sentences express that said person was feeling under the weather and is now getting better.

To explain that you are getting better you can say things like:

I'm getting better.
I'm on the mend.
I'm improving all the time, thanks for asking.