The Trumputin Axis of power
Trump and Putin rely on each other for the legitimacy of their power. The reasons differ but the point is the same. Without each other, neither of them stand a chance.
For a leader to have a mandate they have to be considered legitimate domestically and abroad. Either of these factors may be enough for them to retain power, but without either one they are quickly replaced, either through democratic means or by palace or military coups de tat.
First let's explore how Putin legitimizes Trump.
Trump's treatment of Putin and Ukraine does not make him popular domestically or abroad, but the USA is a vital element of Nato. Without US weapons,US support of Nato, and US sanctions, the alliance is considerably weaker. If European power is weakened, that means the relative power of Russia is increased. This is why European leaders have been so careful to critisize the Trump administration or take more bold moves to push back at Trump. The appeasement is because of dependence. In this way the continuation of the war and the success of Russia gives Trump legitimacy. If Trump ends the war in terms favourable to Russia, Europe will not have time to shore up it's military power and economic resilience for a later attack that is almost certain to happen once Putin regroups and mounts the next attack. If however Putin falls and there is a sustainable peace with a sovereign Ukraine that defeats Putin's war aims, Europe regains its time to prepare and therefore its bargaining power with the US and Trump. European leaders would be much less likely to defer to Trump or heap praise on him.
Putin's reliance on Trump
For Putin, his domestic legitimacy is lowering because the war has not achieved any of its strategic goals in four years. The Russian economy is moving into recession, the national wealth fund is depleting fast, the armor stockpile from before the war is dwindling, domestic problems are piling up and repression at home is intensifying to keep control of the population. However a picture is worth a thousand words. When the president of the most powerful country on earth places a photo of himself and Putin prominently in the inner sanctum of the seat of power the white house, it gives Putin domestic legitimacy.
Having the president of the United States wrapped around his finger is Putin's only real remaining power. He has also lost legitimacy among allies and neighbouring countries. Venezuela, Cuba, Syria and Iran were all massive blows to Putin's power. It may all be worth it if he has the US president as his fanboy, but without it he would be left with no leverage. No cards to play. Domestically Russians would start wondering if Putin's war, and himself are worth the burden. The Logical answer is NO.
If Trump is removed, Putin's power will diminish greatly and make him extremely weak. If he no longer has influence abroad and he his domestically unsusccessful the war will change shape considerably, and with a supportive US administration Ukraine can win. What happens to Putin then is a loss of power at least, and much worse most probably. Russia will be forced to subjugate itself to China or reproachment with the west.
On the other hand, if Putin is deposed Trump will no longer have leverage over allies. This leverage has been reducing but it shouldn't be understated. Without Europe having to keep Trump on side, they can act more decisively on the domestic front without having to keep Trump's friends happy, and they no longer have to worry about being abandoned by the US.
Putin's defeat is Trumps defeat, and Trump's defeat is Putin's defeat. How do ordinary US citizens defeat Trump. Send help to Ukraine. Raise funds for Ukraine like it matters for the existence of their democracy, because it does. For europeans, they need to give Ukraine everything they can to enable them to win the war. Not end the war, not freeze the war, but to push back against Russia and force them into retreat. At this point a massive battlefield loss for Russia would be catastrophic. Ukraine doesn't need to retake all their territory. Even if the victory is symbolic, like blowing up the Kirch bridge, Putin will find that he has no more support domestically at all. This is because besides his influence on Trump, the only thing Russia has going for it is progress on the front lines.
US and European people can find a way out of this mess if they destroy the Trumputin axis of power. The best strategy is to pursue both avenues, deposing Trump and defeating Putin.