No processor that I know of will throttle or increase speeds depending on the PSU. If your PSU is insufficient, you'll get stability problems (crashing).
If you don't want your CPU to "go into turbo mode" go to Control Panel - Power Management Settings - Advanced Settings - Processor Power Management and adjust the "maximum Processor state" to 95%. There may be more graceful ways to accomplish the same thing (disable TurboCore/Turbo Boost in BIOS if the machine supports changes to such values), but what I mentioned should work for all desktops AND laptops.
Main: i5-3570K, ASRock Z77 Pro4-M, MSI RX480 8G, 500GB Crucial BX100, 2 TB Samsung EcoGreen F4, 16GB 1600MHz G.Skill @1.25V, EVGA 550-G2, Silverstone PS07B
HTPC: A8-5600K, MSI FM2-A75IA-E53, 4TB Seagate SSHD, 8GB 1866MHz G.Skill, Crosley D-25 Case Mod