My MR2 is getting a lot of pressure built up in the cooling system. It pushes the overflow tank's water out through the overflow hose. i replaced the thermostat and i keep adding water into the radiator cap. It pumps water but it keeps over heating

no year car told, age matters on all machines. all.
all cars run about 15psi , pure physics of hot water expansion and cap pressure regulated.
my guess is you mean the overflow tank on side of RAD,that has exploding steam and water to this side tank? "violently" that be gross overheating
lots of causes and lots of tests. to find it.

• bad thermostat ,10 years is like max. life.
• dead water pump.
• bad radiator
• bad RAD fan. dead. or dual fan 2nd is dead.
• low coolant, sure,after now but is no cure filled and tried. full up in rad dead cold 50%AF MIX.
• FANS BELT MISSING OR LOOSE. (TO PUMP)
• BLOWN HEAD GASKET OR WARPED HEAD(S) CAUSED.
• SIDE TANK HOSES BAD MORE MISSING SUCH THE EVERY COOL DOWN AF DOES NOT GET SUCKED BACK INTO RAD. MAKING IT DROP LEVELS OVER TIME.
• ALL WATER HOSES CAN LEAk or heater core. causing water loss and bingo overheats.

Doesn't sound like it ran hot enough or long enough to crack or warp the head. If there is a crack, the engine will continue to overheat. If a warped head, car would probably run rough, the affected cylinders would likely have little or no compression, there would be a noticeable miss, and overheating would continue.
Why the low coolant? It's a sealed system, so should never have to add coolant, other than a small loss from evaporation in the overflow tank. Monitor the level closely for the next few days. Note the level in the overflow tank to see if it drops over time. Leaks can be external anywhere in the coolant system-engine, radiator, hoses, heater core- or internal in the engine-head gasket failures can cause coolant in the cylinders (white smoke out the exhaust), or coolant in the oil (oil will have a milky brown look to it), or oil in the coolant (will see traces of oil in the radiator).
A couple of tests may be called for. A pressure test of the coolant system will check if the system holds pressure as it should-about 15 psi. If pressure does not hold, there is a leak somewhere. Water pumps usually leak through the weep hole when they fail. A radiator leak would show up when pressure is applied. If pressure does hold, check if you have a good radiator cap-the cap is what seals the system so pressure will maintain. Coolant under pressure raises it's boiling point by about 10 degrees, so is critical to keep the system under pressure.
If you suspect further problems-losing coolant or continued overheating-then have the chemical test done to check for exhaust gasses in the coolant. Basically a check if the head gasket is good, a simple test done at the radiator cap opening with a special fluid that changes color if hydrocarbons are present in the cooling system.
Or you could have a compression test of the cylinders to check the internal mechanical condition-not only tells you if head gasket is good, also tells you a lot about engine condition-even,. balanced, good compression in all cylinders translates into good power output-rings are good, valves are sealing good, all systems go.
The cooling system: water pump, thermostat, radiator, hoses, heater core and hoses, radiator cap, overflow tank, and the cooling fans and associated wiring. Make sure everything is working right and you'll be good to go.

If the engine overheats from a thermostat
stuck closed or bad head gasket

Answer could be a faulty thermostat,or it could be a sign that your head gasket is about to blow ,or last of all your timing could be out of sync.

Answer 2 from Inventus: It means your cooling system is funtioning properly. In a system having a coolant recovery tank, coolant in the radiator is always up to the brim, hot or cold. There is negligible or no air space. When coolant in the radiator expands sufficiently due to warming from the engine, it will squeeze past the pressure cap's bottom seal and flow into the recovery tank. (If no provision for such expansion was present, the expansion would rupture the radiator or your hoses.) Only coolant within the radiator is under pressure, and because of this pressure (together with the elevated boiling point that the "anti-freeze" permits), it normally does not boil. But once past the pressure cap's bottom seal, the overflow is at atmospheric pressure and therefore boils.
This boiling is usually unnoticed after a short, i.e., local, trip because the cooler coolant already in the recovery tank quenches it. But after some highway driving the influx of more hot overflow heats up all the coolant in the recovery tank to the (unpressurized, i.e., "natural") boiling point.
As the engine cools when shut off, the contracting coolant in the radiator sucks back coolant from the recovery tank. Fluid in the recovery tank should never be below the "full hot" or "full cold" marks, lest air be sucked in.
-- BETTER ANSWER ==
Your cooling fans are not turning on. It is not normal for your overflow tank to boil like that. It is true that your radiator is overflowing into the reserve tank, but that means yourr adiaotor is boiling. Check for blown fuses or relays for your cooling fans. IF theya re fine. run your engine for about 15 minutes and drive. When you temp level is at normal operating temp open your hood with the engine runing and see if your fans are on. If they are, then you may have a bad thermostat or a plugged radiator, or a bad water pump. If the fans are not on, get your cooling fan switch replaced if your car has one. Check your temp sending sensor

You still have a lot of air in the cooling system and you have to speed up the water pump to get any coolant to flow through the heater. Let the engine cool off, then open the radiator cap and top off the radiator. Pour the coolant in at a rate slow enough to let the air bubble up without splashing the coolant back out. Once you get the coolant level up to the top overflow hose, the remaining air will be pushed out by expansion through the overflow tank once the engine gets hot, and will be replaced by coolant from the tank on cool-down.

Due to the complexity of the newer cars on the raod today!! what you are experiencing is Normal!!! since the coolant systems on the newer cars run at such a higher pressure and a high Rpm the reserve tank which has it's on drain hose for excessive overflow from the radiator allows the extra high pressure release from the radiator to back flow to the reserve tank and then out the reserve tank overflow hose on to the raod

about 100K miles? sounds like a blown head gasket is allowing coolant to be pushed out the exaust,and combustion pressure is pushing fluid out the overflow.have your compression tested.

That's what it sounds like, a blown head gasket. Every time the piston compresses it pushes the air fuel mixture into your cooling system through the broken head gasket. Try running the engine with the radiator cap off or lose and look and smell for gas in your radiator overflow tank. Good luck and hope this helps.

yes, not having an overflow reservoir for your radiator will cause you to lose water. As your engine warms up the water expands and creates pressure in the cooling system. When the pressure reaches a certain point(usually about 15 psi.) the radiator releives the pressure by releasing water into the overflow tank. When the engine cools back down the water shrinks pulling water back into the radiator from the overflow. If there is no overflow, the everytime the engine warms up on a drive water will be lost and not recovered on cooldown.

The best way to flush it would be to take off the bottom radiator hose and pump water through the fill nozzle after all the coolent is out you can then hook the hose back up and replace it with new coolant.