There is no contradiction. I'll put it mathematically for you. Do you understand Pythagoras? Given a right triangle with the hypotenuse = c, base = a height = b and "^2" to mean "raised to the second power" then:

a^2 + b^2 = c^2. Both a and b are components of hypotenuse c.
Now do the same to space-time but I will include a mistake only for a temporary illustration:

space^2 + time^2 = space-time^2. Both space and time are components of space-time.

The mistake I made deals with the fact that I failed to include the momentum through each of the 3 spatial dimensions and to further place an important minus sign in front of time^2. That is important to keep the speed of light constant to all inertial reference frames.

You cannot travel through time at speed c unless you fail to displace through space. Even sitting in front of your monitor you have displacement through space in the reference frame of the earth.

You need to understand that there are clocks, many of them all over a reference frame. All of those clocks make up the "observer".They do not move with respect to one another and information from one clock to another does not arrive instantly at another clock.

You further need to study simultaneous events. Without doing that the study of time will not make much sense. What is simulataneous to observers in one frame of reference is not simultaneous in another frame of reference unless the two events take place at the same location.

Mostly, relativity is new to you and you are doing the same thing we all did when starting. You are jumping to many false conclusions and need to really slow things down.

http://www.black-holes.org/numrel1.html