Tell us your mobo maker and exact model number, so we can look up its manual and advise using its features.
You CAN connect a 3-pin fan to a 4-pin header, am discussed above. Just to verify that this has been done correctly, look at the mobo male (with pins) header. The plastic tongue sticking up beside it is beside Pins 1 to 3, and Pin #4 is beyond the tongue. Pin #4 is the one that should NOT be used by the 3-pin fan connector.
Most mobos today use only 4-pin headers, BUT allow the user to change HOW the electrical signals for that header are sent out, through option settings in BIOS Setup for that header. Here we are dealing with the CPU_FAN header. The options often include: Voltage Control Mode (aka DC Mode), PWM Mode, and Automatic. Many are shipped set to PWM Mode or Auto by default. Auto is supposed to test the connected fan at start-up and choose the correct option based on that. But if there is no AUTO option this cannot be done and you need to set the option yourself. Further, some mobos do not even have an option for this header - they use only the PWM Mode and offer no choice.
Now, if you connect a 3-pin fan properly to a header that is using the new PWM Mode (whether by option or by the no-choice-available way), that fan will always run full speed. You report the fan starts, but quickly stops, so your header must NOT be using PWM Mode. Either that, or you have plugged it in wrong. So, MAYBE your CPU_FAN header already is set to use DC Mode (or Auto, but it sets itself to DC Mode). What else could be involved?
Three configuration options for that header might be involved. Some headers have an option for the MINIMUM speed of the fan that you can specify. IF that is set too low for your fan, once it goes into the full control by the header and is told to drop to minimum speed for the cold system, it will stall. The second related item is a choice in the fan PROFILE setting. There the default automatic settings establish the "curve" of what speed to run the fan for what measured CPU internal temperature. Alternatively, you may have the option to set your own custom "curve" using a Profile called Manual or some such. In either of those two cases, you may have the lowest speed (at lowest temperature) set too low and that might need adjustment. And another: the Profile items also include an option for Quiet operations you might have chosen. This is just a setting of a fixed low speed, and that MIGHT be too low.
As I said, If we knew your mobo, we could check its manual and provide more precise advice.