I have heard one scholar from Ahlu-Sunnah Wal-Jamaa that the Tragedy of Karbala indeed took place in the month of Safar and not in Muharram. Also; Its a well-known fact that Husain left for Koofa from Makkah a day or two before Arafat and that means he reached on the outskirts of Koofa in a record time of less than one month. And By all historical records...it becomes clear that the minimum time a person riding the fastest horse to reach Koofa from Makkah (950 miles) would be around two months. Is it possible that Husain could have reached Koofa in less than a month when he was traveling with all his family and his children and his brothers and their families and his Ansar? Please try to reply with authentic Sunni sources.
All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger, may Allah exalt his mention as well as that of his family and all his companions.
It is well-known for the scholars of Ahlus-Sunnah, may Allah have mercy on them, that Al-Husayn Ibn 'Ali, may Allah be pleased with him, was killed on the day of 'Aashooraa', the 10th of Muharram; this is what many scholars, have stated like Al-Waaqidi, Abu Na'eem, Al-Fadhl Ibn Dakeen, Ibn Abu Shaybah, Khaleefah Ibn Khayyaat, Al-Bukhari [in At-Taareekh Al-Kabeer], Ibn 'Asaakir [in the history of Damascus], Ibn 'Abdul-Barr and Al-Khateeb Al-Baghdaadi, may Allah have mercy on them. This is also the authoritative opinion of many late scholars like Ath-Thahabi, Ibn Katheer, Ibn Hajar and others, may Allah have mercy on them. Some of them stated that Al-Hasan Ibn 'Ali, may Allah be pleased with him, was killed in the month of Safar; this is what Al-Waaqidi reported from Muhammad Ibn Ka'b Al-Quradhi, may Allah have mercy on them, and then refuted this saying: 'It is confirmed that he was killed on the Day of 'Aashooraa' (10th, Muharram).' Al-Qurtubi, may Allah have mercy on him, reported that there is a consensus on this and said: 'The scholars, may Allah have mercy on them, agreed in a consensus that he was killed on the Day of 'Aashooraa' (10th Muharram) in the year 61 A.H.'
They stated that he travelled from Makkah to Iraq on the 8th of Muharram. Based on this, he travelled the distance in one month and two days and not in less than a month as stated by the questioner; this is a sufficient period of time for travelling this distance.
For more details on the view of Ahlus-Sunnah about the tragedy of Karbalaa' please refer to Fataawa 87406 and 84152.
Allah Knows best.