"Careful watch" means to closely and attentively observe or monitor someone or something, paying attention to even the smallest details. Full definition
Their behavior may be altered by stress and anxiety, so keep careful watch on them and separate particularly anxious individuals to a safe space if necessary. (country-oaks.vet)
In a previous report, experts said that the home buyers were keeping careful watch on the market and the economy and were maintaining caution before investing in property. (realtytoday.com)
When a baby sleeps next to her parents, it reduces the chances of any accident as parents are able to keep a constant and careful watch over the baby. (parenthood.guide)