The curvature of spacetime is defined by distortions of distance and time measurements. The curvature does not bend spacetime in a 5th dimension.
This is often illustrated by considering the surface of the Earth as an example of a curved space. It is true that the 2 dimensional surface of the Earth does bend in the 3rd dimension. So in this regard, the example is misleading. But in other ways, it is a good example.
Suppose the surface of the Earth was flat, like a paper map. You could go from one corner of a square to the opposite corner two ways, north then east, or east then north, and wind up at the same point.
That doesn't work on the Earth. It is easier to see if you use a big square. Start on the equator. Go north to the pole. Turn left and go south to the equator. Try it the other way. Go 1/4 of the way around the world to the east. Turn right and go to the pole. The two routes lead to two widely different points.
Arriving at two different points is the hallmark of curvature.
GR says that spacetime is curved near matter (Actually near energy density, pressure, and shear stress.) One effect of this is that time runs slower in a gravity well.
A "square" in spacetime can have one distance side and one time side. Suppose you are above a neutron star. The other corner of the square is the place and time straight below you on the surface, one second from now.
There are two routes. You can find the point on the surface below you right now, and then wait one (slowed) second. Or you can wait a second, and then find the point that is right below you at that time. The two routes lead to different place-and-times.
There are also purely spatial distortions. That is, there are distortions in spacetime where all sides of the square are distances. If you wanted to orbit the neutron star, you fight first want to measure the circumference of the orbit. You might also calculate the distance to the center of the neutron star from $2\pi r$. If you measure the distance to the center, you would find it is farther than $2\pi r$.
Reimann developed the mathematics of curved space. But he only considered space where all dimensions are distance.
A prime difference between classical physics and relativity is that time is the same for everybody in classical physics. In relativity, time has some properties in common with space.
I am at rest. My position at time $t_0$ and $t_1$ is the same, $x_0$. You see me as moving. You say my positions at $t_0$ and $t_1$ are different. This confuses nobody. It is a simple effect of motion.
Time has this same property. I am at rest. I am holding two lights at $x_0$ and $x_1$. The lights flash at the same time, $t_0$.
You are moving. You say the lights flashed at different times. This too is a simple effect of motion. It ifs very confusing, but true.
Einstein saw the deep connection. Time behaves so much like space that it must be included in the distance vector. Leaving it out gives wrong answers. It would be like measuring the distance from the base of a mountain to the top on a map. You get the wrong answer if you leave out altitude. This led to SR.
To develop GR, Einstein applied Reimann's mathematics of curved space to spacetime.