The Quran seems to sanction not only child marriage, but also sex with child brides. The following verse talks about "Iddah" or the waiting period that men have to observe before separating from women after claiming divorce:
65:4 - As for your women who have despaired of further menstruating, if you are in doubt, their period shall be three months; and those who have not menstruated as yet. And those who are with child, their term is when they bring forth their burden. Whoso fears God, God will appoint for him, of His command, easiness.
What this verse immediately shows is that the Quran talks about divorce period for immature girls who haven't menstruated yet, clearly sanctioning child marriage. A closer examination of this verse together with the following verse shows that Quran not only approves marriage, but also sex with these girls before puberty. Let's look at this verse:
33:49 - O believers, when you marry believing women and then divorce them before you touch them, you have no period to reckon against them; so make provision for them, and set them free with kindliness.
Thus, it is clear that the concept of Iddah or waiting period arises only if sexual intercourse took place. And yet, the Quran clearly talks about an Iddah for sexually immature girls.
The Tafsirs show the link between these verses - Jalalayn, Ibn-Abbas, Al-Wahidi, Ibn-Kathir - all state that verse 65:4 was revealed because some people specifically had questions about the Iddah for sexually immature girls (and menopausal and pregnant women) who were not mentioned in chapter 33 of the Quran which talks about Iddah. They reference the following hadith:
Ubayy ibn Ka‘b said: ‘O Messenger of Allah, some women of Medina are saying: there are other women who have not been mentioned!’ He asked him: ‘And who are they?’ He said: ‘Those who are too young [such that they have not started menstruating yet], those who are too old [whose menstruation has stopped] and those who are pregnant’. And so this verse (And for such of your women as despair of menstruation…) was revealed”.
To sum it up, it is clear from the Quran alone that sex with prepubescent girls is approved in Islam. The Tafsirs reinforce this claim and show that the verse was revealed specifically to answer the doubts of Muslims including the Iddah for sexually immature girls.