This relationship is determined from ice cores drilled in Greenland and Antarctica. The graph shows two Antarctic core results, clearly showing temperature leading CO2 content. The interglacial or warm period between periods of glaciation shown is the previous one, named the Eemian. It started 132,000 years ago, then ended 118,000 years ago. Not only did CO2 lag the temperature rise, its high level did not prevent the drop back into a new period of glaciation.
It won't prevent this interglacial period from dropping into another glacial period either. It might be noted this interglacial is aging and another glacial period may be near. We might well wish CO2 did have the capability of maintaining warmth.
In the meantime, we might better spend the tens of billions of dollars now expended to prevent global warming by control of CO2 in technology to grow food and keep warm during an ice age -- economically. The need is inevitable.