The family of actress Kim Sae-ron -- who died by suicide in February -- is firing off at a Korean screen star who they claim groomed their daughter ... revealing texts he allegedly sent her when he was an adult and she was still underage.

The family's attorney held a press conference Thursday ... sharing messages allegedly sent in 2016 between 15-year-old Kim Sae-ron and Kim Soo-hyun, who was 28 when the lawyer claims the messages were sent.

Written in Korean, these texts translate to seemingly inappropriate messages between the two ... with Kim Soo-hyun saying he wants to kiss and hug Kim Sae-ron through the night.

Kim Sae-ron's family originally did not plan to release these text messages, their attorney says ... however, the family's trying to put an end to claims made by YouTubers that Kim Sae-ron's relationship with Kim Soo-hyun was a hoax.

In one alleged exchange in June 2016 -- just before her 16th birthday -- Kim Sae-ron wrote "♡kiss," prompting Kim Soo-hyun to respond, "Do it for real later. Is this forbidden, too?"

Kim Sae-ron answered, "No, it's not forbidden. I'll do it."

According to reports ... Kim Sae-ron allegedly wrote a letter to Kim Soo-hyun last year, saying "We were together for five to six years. You were my first and last love. I hope you won't avoid me." That letter, written when she was 23, was never delivered ... it reportedly included a photo of self-harm.

Kim Soo-hyun and his team denied the allegations in a statement to the Korea Times ... saying he only began dating after Kim Sae-ron turned 18.

Kim Sae-ron was reportedly also under pressure to repay a nearly $500K loan to Kim Soo-hyun's agency, Gold Medalist.

As we told you ... Kim Sae-ron was found dead in her home in Seoul back in February -- and, cops ultimately ruled the death a suicide.

Kim Sae-ron appeared in flicks like "The Man from Nowhere" and "The Neighbor." Kim Soo-hyun has starred in movies like "Real" and shows like "Queen of Tears."