Around 30,000 security forces and allied fighters launched Monday the biggest anti-IS ground operation yet in Iraq, closing in on Tikrit from at least three directions.
A senior commander said operations were focused on cutting supply lines of weapons and reinforcements to the jihadists, who seized the city since June.
The next step will be to "surround the towns completely, suffocate them and then pounce on them," Lieutenant General Abdel Amir al-Zaidi told AFP.
On Wednesday, the United States warned the offensive must not fuel sectarian tensions.
"It is important... that this operation should not be used as an excuse or as cover for individuals taking sectarian-motivated retribution," said White House spokesman Josh Earnest.
"That would tear at the fabric of the country, and weaken the ability of the Iraqis to confront this threat to their country."
Troops have still not retaken Ad-Dawr to the south and Al-Alam to the north, but some units were already on the edge of the city, military sources said.
Zaidi said the operation had already secured areas further out in Salaheddin province and was forcing IS fighters to regroup in urban areas.
"The first phase of the battle to liberate Salaheddin was successfully completed -- and in record time -- by clearing the areas in the east of the province," he said.
The government advance has been slowed by car bombs, roadside bombs and sniper fire, as IS fighters retreated to urban positions but seemed unable to fight back in open areas.
- Revenge killings -
Government forces, Shiite militias and volunteer units have been supported by Iraqi jets and helicopters, as well as Iran.
"This is the most overt conduct of Iranian support," General Martin Dempsey, chairman of the US Joint Chiefs of Staff, said Tuesday.
"Frankly, it would only be a problem if it resulted in sectarianism."
But sectarian-fuelled revenge killings have been a feature of past operations and rights groups expressed concern Wednesday.
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"We are concerned about the possible recurrence and increase of such attacks in the ongoing operations," Amnesty International's senior crisis response adviser Donatella Rovera told AFP.
Some leaders and fighters have described the operation as an opportunity to avenge last June's IS massacre of hundreds of new, mostly Shiite, recruits from the nearby base of Speicher.
Some Sunni tribes have been accused of taking part in the massacre, considered the worst of its kind since IS swept through Iraq's Sunni heartland and beyond the same month.
Prime Minister Haider al-Abadi said Sunday when he announced the Tikrit operation that residents should turn on IS.
Speaking to parliament the next day, he said that "in this battle, there is no neutral party," arguing that anyone choosing neutrality was effectively siding with IS.
"Abadi's statement that there can be no neutrality is worrying," said Joe Stork, deputy Middle East and North Africa director at Human Rights Watch.
- Smugglers -
Many civilians fled the cities conquered by IS last year but the group has recently prevented residents from leaving in some cases.
A former army officer who gave his name as Abu Ahmad fled the town of Al-Alam with his wife and five children Sunday and said he had to pay a smuggler.
"We left with a 'guide', a guy who knows the roads. We were five families, and paid him $1,000 (900 euros) each," he said by telephone from Kirkuk.
The Kurdish-controlled city is only 100 kilometres (62 miles) away but the route they took was a huge loop through the desert that saw them cover eight times that distance.
Tikrit, a Sunni city 160 kilometres (100 miles) north of Baghdad on the Tigris River, is of both strategic and symbolic importance in Baghdad's fight against IS.
It is the hometown of late president Saddam Hussein, the remnants of whose Baath party have collaborated with IS.
Commanders have also said Tikrit is a stepping stone towards an even more ambitious operation of retaking second city Mosul to the north, which has been IS's main Iraqi hub.
Defence Minister Khaled al-Obeidi said after meeting his Turkish counterpart, Ismet Yilmaz, that Ankara had expressed "total willingness to help Iraq in all fields, whether in training, arming or equipment."
Turkish cargo planes delivered military equipment to Baghdad on Tuesday, the latest sign that Ankara-- once accused of allowing IS to operate freely on its soil -- was getting more involved in the fight against it.
Speaking in Baghdad, Yilmaz claimed that "after recent intelligence cooperation, Turkey banned close to 10,000 terrorist suspects from entering Turkey. It also expelled 1,200 suspects."