Drawing from two circuits with extension cord
I'm very familiar with DC circuits, operation and safety in working with them. AC is new to me. I'm a craftsman and I work in an unfinished garage behind my house, because it's a rental I can't do any of the upgrades I would like, including actually running power to the garage. Currently I use a heavy gauge extension cord running from my kitchen to the garage to power my lights and equipment, it is divided among the devices with a power strip. Unfortunately this limits my amperage draw, I'm usually fine but if I want to run my compressor, mig welder or belt grinder I will usually have to turn off the lights or trip the 20a circuit breaker, and none of those big machines can run together.
I would like to splice two extension cords together so I can draw from two different circuits in the house. I still only need 120v but the extra amperage would greatly improve my working conditions.
So what I'm looking for is a basic rundown on how I could accomplish this such as a wiring diagram, any safety considerations I need to keep in mind, or a better way to achieve my goal (that doesn't involve moving or rewiring the garage).
Thank you very much.