Great question! Have you googled?

It’s a confusing topic .. so curious if anyone here has an answer or opinion!

Exactly what pesticides are you talking about, natural pesticides/herbicides, or manufactured pesticides/herbicides. Natural pesticides/herbicides will always be in the soil, as they are being produced by the natural system.

Here is a study for manufactured pesticides

https://extension.psu.edu/persistence-of-herbicides-in-soil

https://pesticidestewardship.org/water/pesticide-fate/

The chance of getting an amount of pesticides/herbicides that has an effect from the ground is nearly non existent. First if they did not break down, you could not plant a crop the following year.

If you child is walking, crawling, rolling, or laying on the grass, the he will not pick up pesticides/herbicides in his body. If he is eating the soil, the soil that his eating will cause him problems before any pesticides/herbicides will.

If you are trying to protect your child you should not let him eat peaches, and some other fruit as they contain cyanide.

https://pickyourown.org/apple-seeds-cyanide-arsenic.php

I am definitely no expert but as long as you maintain your lawn organically after it is laid I should think anything used on it by the grower will dissipate and dilute pretty quickly. Your child wouldn't be playing on newly laid turf for a few weeks anyway. We don't know where you are situated but if you're going into winter will the child be out on the lawn much for some months?

Alternatively, sow seed but the lawn won’t be usable for a long time.

Organic turf info.