Does CO2 correlate with global temperature long-term?
While natural variations cause short-term ups and downs to the weather, CO2 and global temperatures move together long term.
This is because CO2 in the atmosphere impedes the escape of heat back to space. Humans added 50% more CO2 since the Industrial Revolution by burning fossil fuels.
Over multi-decade timespans, CO2 shows close correlation with global temperature. However, on much shorter timescales (years, months, days), other natural variations in temperature (e.g. El Nino, La Nina) create "noise" on the graph — the up-and-down fluctuations we see. That's why climate trends tend to be expressed in blocks of 30 years.
When viewed over 30 or more years of data, the dominant relationship between human CO2 emissions and warming is clear. Man made CO2 and global temperatures have both steadily increased since humans have begun to burn fossil fuels.