I want to wire my detached garage from a sub panel that supply 240 volts to my AC unit and water pump. My problem is that there is no neutral leg in this panel. I tried bonding my neutral to the ground but that did not work because in addition to having 120 volts when I put my multimeter leads on black and neutral, I also had 120 volts on black and ground. What should I do
You should contact a local licensed electrician. You should not do what you're attempting, as it may be a code violation and/or dangerous.
The circuit supplying your A/C and water pump, is likely sized to only supply those loads. Adding more loads to that circuit will likely overload it, and cause you headaches in the future.
If you want to supply the garage, you'll want to install a properly sized feeder between the main panel and the garage. This may require an additional panel in the garage, and will require properly bonding and grounding in the garage.
If the panel feeder is large enough to supply the garage, and all you need is a neutral. You'll have to run a properly sized neutral, from the main panel to the subpanel. You cannot use the grounding conductor as neutral.
As it sounds like you're a bit over your head, you might want to contact a professional to do the work properly.
As a side note... In a 120/240V single split-phase system, you should measure 120 volts between ground and either of the hot legs. You should also measure 120 volts between either hot and neutral, 240 volts between the hots, and 0 volts between ground and neutral.
First, mixing ground and neutral is right out. They are very different wires with very different purposes. People often confuse them because they see they are bonded at the main panel. They wildly misunderstand the purpose of that bonding.
OK, so some clever person decided to save a wire and set up a 240V only panel. It's also possible the ground he gave you was cheese.
Wire the neutral properly to the subpanel
You could replace the main-subpanel cable with a /3 cable, or if in conduit, run a neutral. Then the subpanel will have a neutral and you can wire it normally. This would be a good time to re-visit the size of the loads sharing that panel, because if you need to upsize the wire and breaker, now's the time.
In a subpanel, neutral and ground must be separated. They must be separated everywhere except that one engineered bonding location.
Because that bonding location is in the main panel, many main panels have all the neutrals and grounds tossed onto the same bus. In fact, many panels are sold without a separate ground bus on the presumption they will be main panels. For subpanel use, they require you purchase an accessory ground bus, and remove the bonding straps/screws which tie the neutral bus to the panel chassis.
If you have one of those, you'll need to go back to a proper electrical supply house who supports that brand of panel, buy a ground bus, and move existing grounds to it. Then remove the bonding screws/straps from the neutral bus, so you have a proper neutral bus there.
Use a transformer
This is a seriously big-boy strategy, and it still doesn't address the question of whether you have enough power at that subpanel to support all its loads.
Since it's an outbuilding, you'll need to install a grounding system (ground rods) anyway. Why not make it a separately derived service? Supply 240V-only from the subpanel, and then use a transformer to derive 120+240 or just plain 120. Such transformers are readily available as cheap as $100 used.
This transformer then feeds a panel. Because this is separately derived (i.e. it is isolated from the supply), this is not a subpanel, it is a main panel and must have the neutral-ground bond. Ground must be locally derived from the local grounding rods at the outbuilding.