When talking about going to funfairs, there are two aspects to be discussed:
Haraam things that are involved in these funfairs, such as free mixing, wanton display of women (tabarruj), and music. If any of these evils or others are present, then it is not permissible to go there.
Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allah have mercy on him) was asked:
Guardians often take their children to places called funfairs in which there are things that are contrary to sharee‘ah such as the wanton display of some women, but the children are very keen to go to these funfairs. What is the Islamic ruling on going there?
These funfairs – as the questioner mentioned – include some evils. If there are any objectionable things in them, then if it is possible for a person to remove these objectionable things, then he is obliged to go there in order to remove them. If he cannot do that, then it is haraam for him to go there and in that case we say: Take your children out to the country and that is sufficient. But if he takes them to the funfair, in which there is mixing and foolish man who try to chat up women, and there are clothes that it is not permissible for a woman to wear, then it is not permissible to go there unless he is able to remove the objectionable things.
End quote from al-Liqa’ al-Shahri, 75/question no. 8
Shaykh ‘Abd-Allah ibn Jibreen (may Allah preserve him) was asked:
Some fathers – may Allah guide them and all the Muslims – take their families, including little children, big girls, minors and women, to places called funfairs, which include rides for young and old and the women ride in front of one another when they are uncovered and wearing their adornments, and many women and girls go there wearing short or see-through clothes and pants, and some of them barely cover the ‘awrah. And they take pictures of one another with their cameras. Some righteous women – who we think are so, but Allah knows best – go to these places and they do not denounce any of these objectionable things. If we advise them that it is not permissible to go to these places, they argue that there is nothing wrong with it and that this is just for fun and leisure. Some of them even regard that as part of a good upbringing and they regard the one who advises them not to do that as being extremist. We hope that you can offer them some advice and direction concerning this matter and explain the bad consequences that may result from it. Thank you and may Allah protect you and take care of you.
I think that it is not permissible to go to these funfairs which involve what is mentioned in the question. That comes under the heading of doing things that are the causes of corruption and inclination towards sin, and training children when they are young to love wanton display, uncovering and mixing with non-mahrams. There is no doubt that if children, male or female, grow up going to these funfairs and mixing with those evildoers, that will result in them getting used to these haraam matters and regarding them as insignificant, believing that they are permissible and not denouncing them in those places or elsewhere. This will make children like behaving in this manner and imitating those evildoers. It cannot be justified by saying that it is for fun or leisure. There are things that can take the place of that, such as going to the countryside and places that are free of non-mahrams, or going to parks and gardens where there is no free mixing, or doing useful things and learning beneficial things at home, reading books of knowledge and Islamic history. That is a kind of innocent fun and leisure which is free of haraam things and obvious losses to one’s religious commitment and worldly interests. And Allah is the One Whose help we seek.
End quote from the Shaykh’s website (question no. 11036).
The second issue is what these funfairs contained of statues and images which is undoubtedly an objectionable thing, and what they contain of images of horses, monkeys or other things that children ride on. These things are still images and statues. Rather what is permissible of children's toys are those that are handled with disrespect and played with, not those that are shaped in the form of an animate being and are respected, guarded and taken care of.
Shaykh Khaalid al-Mushayqih (may Allah preserve him) said:
With regard to these rides that are in the form of images and statues, it seems that they are not permissible, so it is not permissible to allow children to ride on them because of the stern warnings that have been narrated with regard to making images. The Prophet (blessings and peace of Allah be upon him) said to ‘Ali ibn Abi Taalib (may Allah be pleased with him): “Do not leave any image without erasing it or any built-up grave without levelling it.” Narrated by Muslim.
And he (blessings and peace of Allah be upon him) said to ‘Amr ibn ‘Abasah, when he asked him what Allah had sent him with: “… To uphold ties of kinship, break idols and proclaim the oneness of Allah (Tawheed) and not associate anything with Him.” Narrated by Muslim.
And he (blessings and peace of Allah be upon him) said: “Allah has sent me as a mercy to the worlds and my Lord, may He be glorified and exalted, has commanded me to destroy the idols.” Narrated by Ahmad in his Musnad.
So the questioner should not be swayed by his children’s desires. As for the alternative, I say that it is possible to provide them with permissible games at home or elsewhere, to make up for not taking them to these funfairs in which there are these rides in the forms of animate beings and statues. End quote.
And Allah knows best.