It is possible to paint with oils without using turps, using only oils to thin the paint. Turps is popular because it disperses oil paint quickly. The only 'issue' is that the paint will take longer to dry.
If you're worried about the possible harmful effects of inhaling turps, it's certainly worth considering. Another option would be to swap to water-soluble oils, where the issue of turps never arises. Or consider using an acrylic paint with an extended opening time, such as Golden's Open Acrylics.