How to fix when the amplitude modulated signal reaches the peak, the bottom follows too?

Amplitude modulation with one transistor can be done but difficult in resistive circuits.

Adding one more transistor allows the modulating current (in this case, 10 Hz.) an alternate path - you can now approach 100% modulation. In the case shown below, both bases are biased at 2.0V. Their common emitters are a little more than 0.6V lower, near 1.37V.
The 10 Hz. modulating signal has average voltage of zero volts, and peak amplitude very close to 1.37V. At its positive peak, no voltage appears across the 1k resistor...both transistors have no emitter current.
When modulating signal goes to its negative peak (-1.37V), available current is +2.74 ma. This current would divide equally between each transistor (since their average base voltage is the same). However, the carrier signal applied to one base disturbs this balance causing modulation.

The circuit shown is simplified by using many voltage sources. Base bias could be provided by resistive voltage dividers to produce the 2.0V base bias.
The modulating signal at the emitters could be provided by another transistor acting as a current source varying between 0 - 2.74 ma., raising the transistor count to three. Collector voltage of Q2 (V_out) contains both modulated carrier and the modulating low-frequency signal. The small coupling capacitor C1 reduces the amplitude of the modulating component, while passing full AM signal to R2 (waveform at R2 not shown):

Jonk's solution doubles the transistor count to 6 in a fully-balanced expansion of this idea when he suggests a 1496/1596 chip. A NE612 is a simpler version that is internally biased, allowing both modulating signal and carrier signal to be AC coupled with capacitors.

Edit: A simple conceptual AM modulator uses a logic-controlled single-pole, double-throw switch. The carrier signal is a square-wave logic-compatible waveform controlling switch state. In the schematic shown below, a relay is used to control switch state (not practical for fast carrier switching).

For half the time, the switch connects one modulator source to the output, half the time connecting the other source to output. Be aware that both low-frequency sources are the same signal, exactly out-of-phase. Each V1, V2 modulator source must have a DC offset combined with the 10 Hz signal. The peak amplitude of the 10 Hz signal must never exceed DC offset magnitude, else modulation exceeds 100%.

A relay shouldn't be used to switch between sources. Logic-controlled analog FET switches (example:74HC4053) might be used.

^{simulate this circuit – Schematic created using CircuitLab}
It is not clear in the waveform below that a switch type modulator generates a squarish carrier output containing many harmonics. So you end up with not only the desired AM signal at a carrier frequency of 1 kHz., but also spurious AM copies at 3 kHz, 5 kHz, 7 kHz...these should likely be filtered out to achieve a clean output at only 1kHz (plus its wanted nearby sidebands).

How to fix when the amplitude modulated signal reaches the peak, the bottom follows too?

The reason the "bottom follows too" is that the output of your circuit contains both an AM signal and an amplified version of the modulating signal. You need to attenuate the modulating signal component of the output as much as possible, while only attenuating the AM component only a little.

C2 and R5 of your circuit form a high pass filter. To attenuate the modulating signal from the output, the cutoff frequency should be near the carrier frequency.

I have change the values of R2 and R3 accordingly.

^{simulate this circuit – Schematic created using CircuitLab}

A second problem that I encountered with the circuit is that the gain was too high, leading to clipping. The signal at the collector of Q1 needs to look something like this:

However, instead of looking like that, it was clipped at Vcc and 0.7V. That is, the transistor I used went into saturation and cutoff. This caused significant distortion in the output.

So, I reduced the gain by doubling R1.

I also increased the capacitance of C3 so that the frequency of the R1 * C3 pair was closer to that of the modulating frequency. This allows more of the carrier signal to bypass R1.

(The signal voltages were also tuned).

With these changes, the output looks like this:

Not perfect, because it still contains a noticeable component of amplified modulating signal. That can be seen by the fact that the output is still a bit asymmetric about the 0V axis. But it is much closer to a proper AM signal. A second filter stage would improve the output more.

Look here how it should look like: