How do I purify clothes or carpet if a baby urinates on them?

There's a genuine and well-known distinction here based on the baby's sex and diet: for a baby boy who is still exclusively breastfed (not yet eating solid food), the majority Sunni position — based directly on a hadith of the Prophet ﷺ — is that simply sprinkling or splashing water generously over the affected area is sufficient; there's no need to rub, wring, or fully wash it out. For a baby girl's urine, however, the area must be washed properly like any other urine — thoroughly rinsed and, on fabric, wrung out — even while she is still exclusively breastfed.

Once either child begins eating solid food regularly, this concession ends and their urine must be washed the same as an adult's, on both fabric and hard surfaces. For carpet specifically (where you can't wring it out), the practical approach accepted by most scholars is to pour enough water over the spot to thoroughly soak through it, then blot up or extract the water (a wet-dry vacuum or repeated pressing with a clean cloth works well) until no smell, color, or trace of the urine remains — this counts as valid washing even without literal wringing.

Source: hadith distinguishing a breastfed boy's urine from a girl's urine, Sahih al-Bukhari and Sunan Abu Dawud (hadith of Umm Qais bint Mihsan); general fiqh on removing najasah from absorbent surfaces.