JALALABAD — Every few days, bodies are dumped on the outskirts of the eastern Afghan city of Jalalabad. Some have been shot or hanged, some beheaded. Many have handwritten notes stuffed into their pockets, accusing them of being members of Afghanistan's branch of the so-called Islamic State, according to a field report by BBC News.

No-one claims responsibility for the gruesome, extra-judicial killings, but the Taliban are widely assumed to be responsible. IS carried out a suicide bombing in August outside Kabul airport that killed more than 150 people, and is a fierce rival of the Taliban. The two groups are now engaged in a murky and bloody battle. Jalalabad is the frontline.

Afghanistan is now more peaceful, following the end of the Taliban's insurgency. In Jalalabad, however, their forces are facing a near-daily stream of targeted attacks. IS, known locally as "Daesh," is using some of the same hit-and-run tactics that the Taliban so successfully employed against the previous government, including roadside bombs and stealthy assassinations. IS accuses the Taliban of being "apostates" for not being sufficiently hardline; the Taliban dismiss IS as heretical extremists.

In Nangarhar province, home to Jalalabad, the head of the Taliban's intelligence services is Dr Bashir. He has a ferocious reputation. He previously helped to drive IS out of a small stronghold it had established in neighboring Kunar.

Dr Bashir denies any link to the corpses left on display by the roadside, but proudly states his men have arrested dozens of IS members. Many IS fighters who had been imprisoned under the previous government escaped from jail during the chaos surrounding the Taliban takeover

In public, Dr Bashir and the rest of the Taliban play down the threat of IS. They say the war in Afghanistan is finally over, and they are bringing peace and security to Afghanistan. Anything that undermines that narrative is unwelcome.

Dr Bashir goes so far as to claim that IS does not even formally exist in Afghanistan, despite all the evidence to the contrary.

"The name 'Daesh' refers to Syria and Iraq," he says. "There is no miscreant group with the name of 'Daesh' here in Afghanistan."

Instead, he refers to the militants as "a group of traitors who have rebelled against our Islamic government".

In fact, IS is not only formally present in Afghanistan, it has established a specific offshoot or "province" covering the country, "IS-Khorasan" -- using an ancient name for the central Asian region. The group first established its presence in Afghanistan in 2015 and carried out horrific attacks in the following years, but since the Taliban takeover it has launched suicide bombings in areas of the country its militants were never previously seen.

Earlier this month, IS attacked mosques belonging to the Shia minority in the northern city of Kunduz and the Taliban's stronghold of Kandahar.

Dr Bashir, however, insists there is no cause for concern. "We tell the world not to worry," he says. "If a small group of traitors rises up and carries out such attacks, God willing, just as we defeated a coalition of 52 countries on the battlefield... they will be defeated too." Having fought an insurgency war for two decades, Dr Bashir adds, "it is easy for us prevent a guerrilla war".

But fears over the growth of IS are shared by Afghans already exhausted from years of bloodshed as well as neighboring countries and the West. American officials have warned that IS in Afghanistan could develop the capacity to launch attacks abroad within six months to a year.

For the moment, IS does not control any territory in Afghanistan. The group had previously managed to establish bases in both Nangarhar and Kunar provinces, before being driven out by assaults from the Taliban, as well as Afghan army units backed by American airstrikes. The group has just a few thousand fighters compared to around 70,000 Taliban members, who are now equipped with American weapons.

But there are fears IS could end up recruiting some of the other Central Asian and Pakistani foreign fighters believed to be based in the country, as well as disillusioned Taliban members if rival factions develop within the group in the future. The US is hoping to continue using so-called "over the horizon" strikes, launched from outside Afghanistan, to target IS. The Taliban however, are bullish about being able to take on the insurgents alone.

Many members of IS defected to the militants from the Taliban and the Pakistani Taliban, a linked but separate group. "We know them very well, and they know us very well," one Taliban figure tells me, smiling darkly.

In recent days, dozens of IS members have surrendered to Dr Bashir's forces in Nangarhar. One, a former Taliban member, tells us he grew disillusioned after defecting to IS.

Unlike the Taliban, who have repeatedly stressed that their only aim was to establish an "Islamic Emirate" in Afghanistan, IS do have global ambitions, he tells us.

IS "would issue threats to everyone, to the whole world. They wanted to bring their rule to the whole world," he says. But "words are different from actions," he adds. "They are not powerful enough to take control of Afghanistan."

Many Afghans wearily refer to the uptick in IS attacks as the start of a "new game" in the country. In Jalalabad, it's not just the Taliban who are being targeted. Civil society activist Abdul Rahman Mawen was driving home from a wedding earlier this month when gunmen opened fire on his vehicle. His two young sons, aged 10 and 12 cowered in the car as their father was shot dead. IS issued a short statement claiming responsibility.

Speaking from the family's home, his brother, Shad Noor, is despondent. "From the bottom of my heart, when the Taliban took power, we were very happy and optimistic: that corruption, murders, explosions would be eradicated," he says.

"But now we are realising a new phenomenon is being imposed upon us, by the name of Daesh."