It sounds like your pump is not on the valve properly, so instead of inflating the tire you're just pressurizing the inside of your pump.
Does your bike have Schrader valves (the kind you see on a car) or Presta valves (skinnier, with a kind of pointy top)?
If it has Presta valves, make sure the top of the valve is unscrewed.
There's a little nut that you can thread up, and you need to do that before you can put air in to the tire.
Also, try wiggling the pump nozzle around on the valve while you push down on the pump handle, you should be able to hear the change as air starts going in to the tire.
Image of a closed Presta valve: Image of an open Presta valve: Image of a Schrader valve:
bicycles.stackexchange.com/users/12725/stranger
Pressure is the ratio of force to the area over which that force is distributed, so if you want to use less force, solution may be a double-handle pump with a thinner cylinder.
Pump with a large diameter cylinder can pump air faster but needs more force.
Pump with a smaller diameter cylinder will deliver less air with a single stroke, but requires less force to pump, which may be what you need when you inflate high pressure tire.
bicycles.stackexchange.com/users/13103/user3869827
I've had that happen with schrader valves when I don't get the fitting on the valve stem all the way.
There's a little pin that has to be pressed down in order for air to get into the tire, and if it isn't pressed down enough, it won't allow any air in no matter how much you pump.
Make sure you press hard enough to fully seat the pump fitting onto the valve.
If that doesn't help, your tire valve could be stuck closed, or the pump fitting (the thing you press over the tire valve) might be broken.
bicycles.stackexchange.com/users/13143/George_San_Jose
Inside the pump that attaches to the stem is a micro bolt with a vertical line.
When you attach the pump and pull the lever up to lock the pump to the stem, that little bolt turns and locks the stem to the pump.
Most often I have been in such a hurry that I jam the pump to the stem and pull the lever, not locking the stem, and of course after 60 psi or so it pops off.
The trick is to make sure that vertical line on the stem of the air valve, is exactly lined up with vertical line in air pump connection.
When you pull the lever up very gently you will feel it lock in place.
bicycles.stackexchange.com/users/21357/Lear
Here is the answer for Dunlop valves (= Woods valve, = english valve) .
This kind of valve is actuated by pressure only .
And the modern form uses a small steel ball that is pressed into a rubber ring by the pressure in the tire.
The trouble is, that the ball gets quite comfortable within that ring, and it will take some force to push it back out. Force that must be applied in addition to the force needed to overcome the pressure in the tube .
The effect is, that when you want to pump a tire that's currently at 2 bar, you may need to apply a whopping 5 bars pressure from the pump, just to open the valve.
You will feel/hear when the ball comes free (because that's when the content of your pump tube suddenly rushes into the tire).
After that, pumping is easy and straight forward, but you must deliver this one first pressure peak first.
If you already have 5 bar within your tire, it may be plain impossible to free the valve.
I had that several times.
I actually unscrewed the valve in these cases to deflate the tire entirely, then screwed it back tight, and only then was I able to blow the valve free with my pump.
Ps
: I'm switching to Schrader valves now, they don't have this issue.
bicycles.stackexchange.com/users/36555/cmaster - reinstate monica
Air is not going in valve.
If Presta valve, the nut on top must be loosened.
Then tap the end with your finger.
You'll hear a little air escape.
Connect the pump chuck and inflate.
If a Scrader valve (like a car tire valve), the chuck probably is not pressed on far enough.
bicycles.stackexchange.com/users/11419/Malarky
Even after years this was helpful.
I stuck a screw driver into the valve to let some air out.
That loosened up the valve enough for the pump to engage.
If you are trying to find out what is wrong with the pump, don't forget to check whether the problem is with the valve instead.
bicycles.stackexchange.com/users/44238/Bill
When trying to inflate my bike tire, I pump maybe 5 times then it gets difficult for me to pump because the handle is hard to pump?
Difficulty inflating my bike tire
bicycles.stackexchange.com
<tire><inflation>
bicycles.stackexchange.com/questions/23813
bicycles.stackexchange.com/users/13102/user13102
It sounds like your pump is not on the valve properly, so instead of inflating the tire you're just pressurizing the inside of your pump.  If it has Presta valves, make sure the top of the valve is unscrewed and if a Scrader valve (like a car tire valve), the chuck probably is not pressed on far enough. This happens a lot with schrader valves when the fitting on the valve stem isn't all the way. Make sure you press hard enough to fully seat the pump fitting onto the valve.  The trick is to make sure that vertical line on the stem of the air valve, is exactly lined up with vertical line in air pump connection. The pump is not entering the tyre, and is only pressurising the pump itself. The valve may be stuck or not properly opened, or the pump may not be properly seated on the valve. On Schrader valves, the spring can be released with a screwdriver, and on Presta valves the pin needs to be unscrewed properly. A pump with a smaller diameter cylinder may help by delivering less air per stroke, but with more pressure.
The pump is not entering the tyre, and is only pressurising the pump itself. The valve may be stuck or not properly opened, or the pump may not be properly seated on the valve. On Schrader valves, the spring can be released with a screwdriver, and on Presta valves the pin needs to be unscrewed properly. A pump with a smaller diameter cylinder may help by delivering less air per stroke, but with more pressure.