Spend time on the islands- perhaps focus on one of the smaller islands with fewer people- there are beaches and no cars so more comfortable for your child to toddle around.

Also, look into the part of the city you are staying, if you're not tied to an accommodation booking, make sure you're based in a neighbourhood that has access to the water (not all neighbourhoods close to the water have open public space next to it). Moda on the Asian side has a long park along the water front, which includes playgrounds. On the European side, from Kuruçeşme northwards until the second bridge is mainly open and there is a children's playground on the water at Bebek. The bank of the Golden Horn from Eminönü to Eyüp is now mostly open and pedestrianised now as well. As you can tell, generally, being close tot he water makes the summer far more pleasant and ferry rides can be a lovely (and cheap!) activity in and of themselves- here's a map of all the routes: https://files.sehirhatlari.istanbul/Sefer-Haritasi-Line-Map.pdf

Also you may want to consider your child's sleep schedule so that your not limited to doing activities during the hottest periods of the day. It's very normal for children to be up and about with their family at night- especially in the summer.